Preheat oven to 350 degrees. Using an electric mixer, cream peanut butter and butter in a mixing bowl until smooth. Add brown sugar; beat until combined, scraping down bowl as needed. Add eggs and vanilla, and beat until incorporated.
In a small bowl, whisk together flour, baking powder, and salt. With mixer on low speed, gradually beat in flour mixture in two batches.
Scoop dough by the tablespoon; roll into balls. Place granulated sugar on a plate; roll balls in sugar, coating completely. Place 2 inches apart on a parchment lined baking sheet.
Bake until cookies begin to puff up slightly, about 7 minutes. Remove from oven. Press one peanut butter cup in center of each cookie. Return to oven; continue baking until cookies are golden brown and chocolate has begun to melt, about 6 minutes more. Let cool at least 10 minutes on baking sheet before transferring cookies to rack to cool completely.