1. Preheat the oven to 350 and place the shelf in the middle of the oven.
2. Spray the inside of a 13×9 pan with nonstick baking spray or liberally butter and flour the pan. Line the bottom with parchment paper. Set aside until ready to use.
3. Whisk the flour, baking powder, and salt together in a bowl to combine. Set aside.
4. Beat the eggs for 1 minute on medium speed with the whisk attachment (6 on a stand/hand mixer).
5. Add the sugar slowly in a stream, raise the speed to medium high (8 on a stand/hand mixer), and beat for 6-7 minutes, scraping the sides down with a rubber spatula once or twice.
6. While you are whipping the eggs, heat the butter, oil, and the milk in a small pot over medium low heat. You can also put the milk, oil, and butter in the microwave and heat on high for 2 minutes if you prefer. (You want the butter to be melted and the milk hot, but not boiling.) Once the butter has melted, it’s basically ready to use (takes about 3-4 minutes). Keep the milk on low heat until you are ready to use it. You need it to be hot, not warmish.
7. Once the eggs are done being whipped, add the flour mixture in thirds. First add ⅓ of the flour and mix on the lowest available speed (Stir or 1 on stand/hand mixer) for about 10 seconds, then add the second ⅓ and mix for 10 seconds, and finally the last ⅓ for another 10 seconds. Turn off the mixer as soon as you see the flour mixed in.
8. Pour the hot milk into a large bowl and add the vanilla. Next add a generous cup of the egg mixture into the hot milk and whisk quickly and briskly until it’s a smooth, thick, and foamy mixture. 
9. Turn the mixer back on speed 1 and slowly pour the milk mixture into the bowl and mix for 10 seconds. Turn the mixer off and take out the whisk.
10. Use a rubber spatula to scrape down the sides and gently fold a couple of times to combine any separate egg batter and milk batters until it is smooth.
11. Pour the batter into the prepared pan.
12. Hold the baking pan by the edges and bang the pan down on the counter 8-10 times (I first put down a wet kitchen towel so it deadens some of the sound) to burst any large bubbles in the batter.
13. Bake for about 20-25 minutes until golden brown and a toothpick testing the middle comes out clean. 
14. Let the cake cool on the counter for 15 minutes before gently turning it out of the pan onto a cooling rack.
15. Let the cake cool completely before frosting. 

For the Frosting:

1. Place the flour, sugar, and milk in a saucepan and whisk until the mixture is smooth and lump free.
2. Put the pan over medium heat and cook, stirring constantly with a whisk, until the mixture has thickened and looks like a custard or pudding, about 12 minutes.
3. Take a tiny taste. There should be no floury or powdery taste. If it does, cook it for another minute or two to cook it out.
4. Remove the pan from the heat, add the vanilla extract and salt, and stir to combine.
5. Then transfer this sweet roux into a bowl. Cover it with plastic wrap, pressing the plastic wrap down onto the surface to prevent a skin from forming. 
6. Let the bowl cool to room temperature (you can speed this up by putting the bowl in the freezer for 20-30 minutes). You can also make this ahead and store it in the fridge until ready to make the frosting. You will need to let the roux warm up to room temperature first before continuing with the frosting. The closer in temperature the roux and the butter are to each other, the easier it will be to whip up a thick and smooth frosting. (Do not use the roux warm as it will melt the butter and you will have a soupy mess. A little cold is fine as it will warm up as it gets beaten.)
7. Place the butter into the mixing bowl with the whisk attachment. Beat the butter on medium high speed (#8 speed on the stand/handheld mixer) for about 3 minutes until the butter is fluffy and is a pale color.
8. Lower the mixer speed to medium (#6 speed on the stand/handheld mixer) and start adding the sweet roux to the butter in generous 2 tablespoon increments. As soon as the roux gets blended in, add in another dollop of roux. (You can turn off the mixer when you’re making the additions if you find yourself making a mess. Then turn the mixer on to #6 speed to incorporate. Switch between on and off until all of the roux has been mixed in.)
9. Once all of the roux is added, raise the speed to #8 (medium high), and whip the frosting for 2 to 3 minutes until the frosting is thick, fluffy, and can hold its shape in peaks. Set it aside while you prep the fruit. 

Prep the Fruit:

1. Wash and dry the strawberries.
2. Slice the strawberries into thick slices. Set aside in a bowl.
3. Wash and dry the blueberries on paper towels. Lay a couple of new sheets of paper towel in a bowl and set the blueberries in the bowl to finish drying and soak up any remaining water.
4. Wash and peel the mangoes.
5. Slice both of the mango into thin slices cutting around the pit.
6. Using a small 1 inch biscuit cutter, punch a circle out of one of the slices and set aside.
7. Next cut out 3 small stars, approximately 1” in size, out of the mango slices, either using a cutter or using a small knife.
8. Cut 8 small strips about 1 inch long and then 16 shorter strips to put around the circle to form the center star.

Assemble the Cake:

1. Cut the cake in half horizontally using a serrated knife.
2. Transfer the bottom piece of cake to a serving platter since it is difficult to transfer the cake once it is frosted and decorated.*
3. Put about 4 cups of frosting on the bottom piece of the cake. Using an offset spatula, spread the frosting smoothly across the surface. Lay the thin slices of mango across the cake evenly.
4. Cover with the top piece of cake. Use another 4 cups of frosting to smooth over the top and sides. Then put the rest of the frosting into a piping bag set with a medium round tip and set aside. You will be piping a border after you place the fruit.
5. Next, find the center of your cake and draw a light mark through the frosting using a chopstick or some utensil. Draw one line horizontally and one line vertically through this center spot. Then draw a line from the top left corner to the center spot. And finally draw a line from the bottom left corner to the center spot. You will see that you have drawn a triangle on the left side that cuts into the two halves of the cake.
6. Now it’s time to decorate with your prepared fruit.
7. At the intersection of the lines, start putting down a row of strawberries on the bottom half of the cake just under the line, on the right side of the triangle. Overlap the strawberries slightly, like shingles. Keep the strawberries just shy of the edge of the cake. You will be filling up the entire wedge section below the triangle on the left side.
8. Now place the three small mango stars at the three corners of the triangle area on the left. In the center of the triangle, place the circle. Next take the eight longer strips and place them around the circle. Finally you will take the sixteen smaller strips and put two strips next to each of the longer strips. 
9. Next you will place the blueberries at the top of the cake so that they mirror the strawberries. Start right above the strawberries and neatly place the blueberries in even lines all the way to the top and edge of the cake. 
10. Pick up the piping bag and push out any air at the open top of the bag and twist it shut. Squeeze from the top of the bag until you see some frosting coming out of the tip. Hold the tip close to the bottom edge of the cake. Squeezing gently, pipe a straight line along the edge. Then do the same at the top edge of the cake. The piping helps the cake look sharper and neater.
11. Take a quick look and admire your work!   
12. Clean up any smudges around the platter and serve your flag cake with pride.