01 - Whisk together flour, baking powder, and salt in a medium bowl until evenly combined.
02 - Beat softened butter and granulated sugar in a large bowl with an electric mixer on medium speed until light and fluffy, approximately 2 to 3 minutes.
03 - Add the egg and vanilla extract to the butter mixture. Beat until fully incorporated and smooth.
04 - Gradually add the flour mixture to the butter mixture, mixing on low speed until just combined. Do not overmix.
05 - Divide dough in half and shape each portion into a flat disc. Wrap tightly in plastic wrap and refrigerate for at least 1 hour until firm.
06 - Preheat oven to 350°F. Line baking sheets with parchment paper.
07 - On a lightly floured surface, roll out chilled dough to 1/4-inch thickness. Cut out rounds or shapes using a 2-inch cookie cutter.
08 - Place cut cookies 1 inch apart on prepared baking sheets.
09 - Bake for 8 to 10 minutes until edges are just beginning to turn golden. Remove from oven and let cool completely on wire racks.
10 - Beat softened butter in a bowl until creamy. Gradually add powdered sugar, then milk and vanilla extract. Beat until light and fluffy.
11 - Divide buttercream into small bowls. Add a small amount of gel food coloring to each bowl, mixing until desired pastel shade is achieved.
12 - Spread or pipe a layer of colored buttercream onto the flat side of half the cookies. Top with remaining cookies to form sandwiches.