Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper and set aside.
Measure 1/4 cup rainbow sprinkles. Reserve 2 tablespoons of the sprinkles for topping and set them aside; use the remaining 2 tablespoons in the dough.
In a large bowl, combine 1 cup unsalted butter (softened) and 1 cup granulated sugar. Using an electric mixer, beat on medium speed until the mixture is light and fluffy, about 2–3 minutes. (If you are using salted butter, omit the 1/2 teaspoon salt listed in the ingredients.)
Add 2 large eggs to the butter–sugar mixture, one at a time, mixing until each is incorporated. Scrape the bowl sides as needed.
Add 1 teaspoon vanilla extract and 1 1/2 teaspoons cake batter flavoring to the wet mixture and mix briefly to combine.
In the same bowl (or a separate bowl), add 2 teaspoons baking powder, 3 cups all-purpose flour, and the 1/2 teaspoon salt (unless you used salted butter). Mix on low speed just until the flour and leavening are incorporated and no large dry streaks remain. Do not overmix.
Fold in the 2 tablespoons of rainbow sprinkles reserved for the dough until evenly distributed.
Portion the dough into 12 equal balls using a 1/4-cup measuring cup (or ~75 g per portion). You may have a very small piece of dough left over.
Place six dough balls on each prepared baking sheet, spacing them at least 2 inches apart. Using the bottom of a glass or your palm, gently press each dough ball to spread it to just under 1/4 inch thickness.
Bake the cookies at 350°F for 11–12 minutes for a slightly underbaked, soft center. If you prefer firmer, more well-done cookies, bake 1–2 minutes longer. Rotate pans front-to-back halfway through baking if your oven runs unevenly.
Remove the baking sheets from the oven and let the cookies rest on the sheets for 2 minutes. Transfer the cookies to a wire cooling rack and cool completely before frosting.
Make the frosting: in a medium bowl, combine 4 ounces softened cream cheese, 3 tablespoons softened butter, 2 cups powdered sugar, 3 tablespoons milk, and 1/2 teaspoon cake batter flavoring. Beat with an electric mixer until smooth and spreadable.
Transfer the frosting to a piping bag fitted with a medium tip (or use a small spatula). Pipe each cookie in a pinwheel pattern, starting from the center and working outward. Immediately sprinkle the reserved 2 tablespoons of rainbow sprinkles over the fresh frosting.
Let the frosted cookies sit for about 30 minutes to allow the frosting to set before serving.