Preheat oven to 350°F. Line 3 (8-inch) round cake pans with parchment paper, spraying with non-stick cooking spray before and after place parchment in pans. Set aside.
Place all of the cake ingredients (except sprinkles) in a large bowl. With an electric mixer, beat on low for 10-15 seconds, then turn up to medium and beat for 2 minutes. Stir in sprinkles by hand. Divide batter equally into the 3 prepared pans. Bake for 30-35 minutes or until a toothpick inserted in middle of cake comes out clean. Let cool.
For the Swiss Meringue Buttercream
Note: You really need an ELECTRIC MIXER for this recipe. Place egg whites and sugar in a large bowl (I used the Kitchenaid bowl that comes with the mixer). Whisk until combined. Place the bowl over a pot of simmering water and whisk like crazy for 2 minutes or until the sugar has dissolved. DO NOT STOP whisking or you will have scramble eggs. Remove from heat and beat on high for 20 minutes with the whisk attachment of your mixer.
Once your egg whites and sugar mixture have been beaten for 20 minutes, you should have stiff peaks of meringue that has just about come to room temperature. Now you're ready for the butter. With the mixer on low, add your butter 2-3 tablespoons at a time. Once you've added all the butter, then add your vanilla extract and flavoring. Mix until combined. Your mixture will be a bit soupy at this point, so you will need to refrigerate for 2 hours or overnight.
After the buttercream has been chilled, take it out of the refrigerator and let it sit for an hour or so. You don't want it TOO cold. I know, this is a bit fussy, but I promise it's worth it. Now you can beat your buttercream again on medium high and watch the magic. At first it might look sort of curdled but don'w worry, just keep beating it. After a few minutes, it will turn into a light and fluffy buttercream. This could take anywhere from 1-4 minutes. Now you're ready to frost your cake.
Assembling cake and decorating
Fill and frost your cooled cake with an offset spatula. Smooth the sides with a bench scraper and a rotating cakestand (if you have it). Let it chill in the fridge while you prepare the colors. Take the remaining frosting and divide into 3 bowls. Color one bowl red and the other blue. Place into large pastry bags fit with a 1M pastry tip. You will need 3 bags and 3 tips. Start with the blue frosting and pipe a sideways shell border in the upper left quarter of the cake to create the blue part of the flag. Then start at the top with red and pipe another sideways shell border horizontally. Then white and keep alternating red and white until you've covered the entire top of the cake. That's it! You did it! Now serve and enjoy. Store cake in the refrigerator for up to 3 days. Serve at room temperature. Note: This buttercream frosting gets very hard when it is chilled but is just perfect at room temperature.
It's a good idea to make the Swiss Meringue Buttercream ONE DAY AHEAD as it should be refrigerated over night for best results.