Prepare the Cookies: Beat butter and sugar in a stand mixer fitted with a paddle attachment on medium speed until light and fluffy, about 1 minute. Add eggs, and beat until just incorporated; beat in vanilla.

Stir together flour, baking powder, baking soda, and salt in a small bowl. Gradually add flour mixture to butter mixture, beating on low speed until just incorporated, about 1 minute. Fold in either assorted red and green sprinkles or assorted blue and white sprinkles until evenly distributed throughout batter. Cover and chill at least 1 hour or up to overnight.

Preheat oven to 325°F. Shape chilled dough into 48 (1-inch) balls; arrange 2 inches apart on 2 baking sheets lined with parchment paper. Working in 2 batches, bake until cookies are lightly browned around edges, 8 to 10 minutes. Transfer cookies to wire racks; cool completely, about 20 minutes.

Prepare the Vanilla Buttercream: Beat butter with an electric mixer fitted with a paddle attachment on medium speed until creamy, about 30 seconds. Add powdered sugar, vanilla, salt, and 1 teaspoon milk, beating on low speed until smooth and creamy. If needed, beat in up to 1 teaspoon remaining milk to reach desired consistency. Store Vanilla Buttercream in an airtight container in the refrigerator up to 2 weeks. 

Spread about 1 tablespoon Vanilla Buttercream onto the flat sides of 24 of the cookies. Top with flat sides of remaining 24 cookies. Roll sides of cookie sandwiches in either assorted red and green sprinkles or assorted blue and white sprinkles (to match the colors used inside cookies). Store in an airtight container up to 1 week.