Step 1

Stir together flour, baking powder, and salt in a medium bowl; set aside. Beat 1 cup of the butter and 4 ounces of the cream cheese with a stand mixer fitted with a paddle attachment on medium speed until creamy, about 2 minutes, stopping to scrape down sides of bowl as needed. Add granulated sugar; beat until light and fluffy, about 2 minutes, stopping to scrape down sides of bowl as needed. Add egg and 2 teaspoons of the vanilla; increase mixer speed to high, and beat until fully combined and fluffy, about 1 minute. Reduce mixer speed to low; slowly add flour mixture, beating until fully combined, 2 to 3 minutes. Cover dough with plastic wrap. Chill until firm, at least 1 hour or up to 3 days.

Step 2

Preheat oven to 350°F with racks in upper third and lower third positions. (If dough chilled longer than 2 hours, let come to room temperature for about 20 minutes.) Line 2 baking sheets with parchment paper. Using a 1/4-cup measuring cup, scoop 18 dough scoops about 3 inches apart on prepared baking sheets. Using bottom of measuring cup, gently press dough scoops to flatten to about 1/2-inch thickness.

Step 3

Bake in preheated oven until cookies are puffed and very lightly browned around edges, 13 to 15 minutes, rotating baking sheets between top and bottom racks halfway through bake time. Let cookies cool on baking sheets 5 minutes. Transfer cookies to a wire rack; let cool completely, about 30 minutes.

Step 4

Beat remaining 1/2 cup butter and 4 ounces cream cheese with a stand mixer fitted with a paddle attachment on medium-high speed until creamy and no lumps remain, about 2 minutes. Reduce mixer speed to low; gradually add powdered sugar, beating until combined, about 1 minute. Add remaining 1 teaspoon vanilla; beat until combined, about 30 seconds. Increase mixer speed to high; beat mixture until fluffy, about 1 minute. (If dyed frosting is desired, divide frosting and food coloring of choice among separate bowls; beat each mixture on medium speed until colors are vibrant, about 1 minute.) Spread about 2 tablespoons frosting onto each cookie; decorate with sprinkles.