Beat butter and sugar in a mixing bowl with an electric mixer until light and fluffy; beat in egg and vanilla extract. Combine flour, baking powder, and salt in a bowl and mix the dry ingredients thoroughly into butter mixture. Drop cookies about 2 inches apart, 1 teaspoon at a time, onto ungreased baking sheets.
Bake in the preheated oven until edges of cookies are golden brown, 12 to 15 minutes. Cool cookies on wire racks.
