In a medium bowl, whisk together flour, baking powder, and salt together until combined. Set aside.
Add butter to a large bowl fitted for a stand mixer. Using the paddle attachment, beat butter on medium until light and fluffy, about 1 minute. Add sugar and beat for 2 more minutes. Add egg and vanilla to mixture and beat until combined.
Add flour mixture to butter mixture just until the dough forms. Separate dough into two halves (I use a food scale to make sure it's exact). Place one half of dough on a piece of parchment paper. Roll out into a 1/4 inch sheet of dough (it doesn't need to be a specific shape). Repeat with other half of dough.
Place one parchment paper with sheet of dough on top of the other and then place on baking sheet. Place in refrigerator for 1-2 hours. If you must chill longer, place plastic wrap on top layer of dough to prevent drying.
When the dough is ready, preheat oven to 350 degrees. Remove one sheet of dough from fridge and place on counter. Using a 2-inch round cutter, cut as many circles as possible. Place circles on a cookie sheet lined with parchment paper about 1 inch apart. They will barely spread so 1 inch is more than enough space between cookies. Gather dough into a ball and roll out again and cut more circles. Repeat until most of the dough is used. You should have about 36 rounds.
Bake cookies in preheated oven for 10 minutes, or until edges of cookie are slightly golden brown. While cookies are baking, take second sheet of dough from fridge and begin cutting rounds and place on baking sheet. Now, we'll cut out the centers to make the tops of the cookies. Using a 1-inch cutter, remove centers from the cut rounds. Gather all the dough scraps and roll them out again and repeat cutting process. You should have 36 tops. Bake cookies for 9 1/2 minutes (the tops take less time to bake).
Remove cookies from oven and let them cool on baking sheet for 5 minutes and then transfer to wire rack to finish cooling completely.