Preheat oven to 325°F and line a 9x13-inch rimmed baking pan with parchment paper, leaving a 1-inch overhang on two sides.

Cream 12 tablespoons unsalted butter and 1 cup granulated sugar together until light and slightly aerated, about 2–3 minutes with a mixer.

Add 1 large egg and 1 tablespoon vanilla extract and mix until just combined.

In a separate bowl whisk 2 1/3 cups all-purpose flour, 1 teaspoon baking powder, and 1/4 teaspoon fine salt, then add to the butter mixture in two additions and mix on low until just combined.

Press the dough evenly into the prepared pan, using an offset spatula or floured fingers to smooth the top, then cover and chill in the refrigerator for 20–30 minutes.

Bake chilled dough at 325°F for 20–26 minutes until edges are faintly golden and the center is set (not jiggly).

Remove the pan to a wire rack and cool completely in the pan, at least 1 hour, before frosting.

Beat 10 tablespoons unsalted butter until smooth, add 3 cups powdered sugar gradually, then add 1 tablespoon vanilla and 1–2 tablespoons milk to reach spreading consistency.

Stir in 1 tablespoon meringue powder (or 1 teaspoon cornstarch) for stability and beat until smooth; divide and tint small portions red and blue if desired.

Spread a thin even layer of white buttercream over cooled bars, pipe or dot red and blue frosting as desired, and sprinkle red, white, and blue sprinkles while frosting is soft.

Chill the frosted pan 20–30 minutes for cleaner cutting, then use a hot, dry knife wiped between cuts to slice into 24 bars.