Preheat the oven to 350°F (177°C). Line two large baking sheets with parchment paper and set them aside.
In a medium bowl, whisk together 1 1/2 cups all-purpose flour, 1 1/4 tablespoons cornstarch, 1 teaspoon baking powder, 1/2 teaspoon baking soda, and 1/4 teaspoon salt. Set the dry mixture aside.
In a stand mixer fitted with the paddle attachment (or in a large bowl with a hand mixer), beat 1/2 cup unsalted butter (room temperature) and 3/4 cup granulated sugar on medium speed until creamy, about 1–2 minutes.
Add 1 large egg and 1 tablespoon vanilla extract to the butter mixture and beat until combined.
With the mixer on low speed, gradually add the flour mixture and beat just until incorporated. Stop once or twice to scrape the sides and bottom of the bowl so everything mixes evenly.
Using a rubber spatula, fold 2/3 cup rainbow sprinkles into the dough until evenly distributed.
Scoop or measure 2 tablespoons of dough, roll into a ball, and place on a prepared baking sheet. Using the bottom of a glass, lightly press each ball to flatten into a thick disk. Repeat with remaining dough, leaving space between cookies on the sheet.
Bake the cookies in the preheated oven for 10–12 minutes, until the edges are slightly golden. The centers may look soft; they will firm as they cool.
Remove the cookies from the oven and let them cool completely on the baking sheet.
While the cookies are cooling (or after they are cool), prepare the cake-mix component for the buttercream: spread 1/4 cup dry vanilla cake mix in an even layer on a parchment-lined baking sheet and bake at 350°F for 5–7 minutes to toast. Remove and let the toasted cake mix cool completely before using.
Make the buttercream: in a stand mixer or large bowl, beat 1 cup unsalted butter (room temperature) until creamy. Add the cooled toasted cake mix, 2 1/2 cups powdered sugar, and 1 teaspoon vanilla extract. Beat on low to combine, then increase speed and beat until the frosting is smooth and spreadable, scraping the bowl as needed.
If using a single color, add a few drops of the desired food coloring (pink, violet, or blue) to the full batch of buttercream and beat until evenly colored. If using multiple colors, divide the buttercream into separate bowls and add a few drops of pink, violet, and/or blue to each bowl as desired, mixing until evenly colored.
Fit a piping bag with a star piping tip, fill it with the colored buttercream, and pipe frosting around the edges of the cooled cookies.
Let the frosted cookies stand at room temperature for about 30 minutes (or chill briefly) for the frosting to set before serving.