Preheat oven to 350°.
Whisk together flour, baking powder, and salt. Set aside.
Using an electric mixer, mix peanut butter and butter until smooth. Add brown sugar and beat until combined. Add eggs and vanilla, mixing until combined. Gradually add flour mixture, mixing just until combined.
Scoop dough by tablespoons and roll into balls. Roll balls in granulated sugar to coat completely. Place 2 inches apart on baking sheets.
Bake about 7 minutes, or until cookies begin to puff slightly. Remove from oven, and press one peanut butter cup in the center of each cookie. Bake for about 6 minutes, or until golden brown and chocolate has begun to melt.
Cool 10 minutes on pans. Then, transfer to wire racks to cool completely.