Cream butter, brown sugar, and granulated sugar on medium speed. Beat in peanut butter, egg, and vanilla until light and smooth. Mix in dry ingredients on low. Chill dough at least 1 hour (up to 3 days).
Preheat oven to 350°F. Take 1 tbsp dough and press a peanut butter cup into a small cradle shape. Top with another 1 tbsp dough and seal around the cup. Roll into a smooth ball. Repeat with remaining dough and cups.
Bake 10–11 minutes until very lightly browned on the sides. Press down any tall cookies. Cool on the sheet 10 minutes before transferring to a rack.
Make the drizzle: melt chocolate chips and peanut butter together, stirring until smooth. Drizzle over cooled cookies.