In a medium bowl, whisk together flour, cocoa powder, and baking powder. Set aside.
In a large bowl, beat butter and sugar until smooth and creamy (about 1–2 minutes).
Add egg and vanilla; mix until fully incorporated.
Gradually add dry ingredients and mix just until the dough comes together.
Shape dough into a disk, wrap, and refrigerate for 20–30 minutes.
Preheat oven to 375°F.
Roll dough to 1/4-inch thickness on a lightly floured surface and cut with cookie cutters.
Place cookies on a baking sheet about 3/4? apart.
Bake 7–9 minutes, just until set and slightly puffy.
Cool completely before frosting.