In a medium bowl, stir together 3 cups flour, 3/4 cup cocoa, 1/2 teaspoon of baking powder, and 1/2 teaspoon of salt. Set aside.
In the bowl of a stand mixer, beat together the 1 cup butter and 1 1/2 cups sugar until creamy.
Add egg and beat until combined. Add vanilla and mix again.
Gradually add the flour mixture to the butter mixture until dough gathers into ball and pulls away from the mixing bowl sides.
Divide dough into 4 equal sections; wrap in plastic wrap and refrigerate for at least 1 hour until dough is firm.
Preheat oven to 350° F. Line several cookie sheets with parchment paper.
Working with one dough section at a time, roll out to about 1/4-inch thickness. Cut into desired shapes with cookie cutters. Transfer to a prepared cookie sheet. Sprinkle with colored sugar or sprinkles, if desired.
Bake 7 to 10 minutes or until edges are just set. The center of the cookies will still be soft.
Remove cookie sheet from oven to wire rack. Allow cookies to cool completely on cookie sheet before removing. They will continue to bake and firm up as they cool.
Repeat rolling and baking cookie dough until all cookies are made.