Heat oven to 350 degrees F. Coat a miniature muffin pan with nonstick cooking spray. Alternately, line the pan with mini paper or foil baking cups. Set aside.
Unwrap the peanut butter cups, place them on a plate or small baking sheet, and transfer to the freezer to chill while the cookies bake. (This helps them hold their shape as you press them into the hot cookie cup).
Add the butter, granulated sugar, brown sugar, and peanut butter to a large mixing bowl. Using an electric mixer, beat at medium speed, scraping bowl often, until creamy. Add the egg and vanilla and continue beating until well mixed. Add the flour, baking soda, and salt and beat at low speed, scraping the bowl often, until well mixed.
Use a small cookie scoop to scoop the dough, then use your hands to shape it into a 1-inch ball. Place the ball into one of the prepared miniature muffin cups. Repeat with the remaining dough to fill the muffin pan. (Balls will look small in the muffin cups but will fill out as they bake and rise to be level with the top of the muffin pan.)
Bake for 12 to 13 minutes or until lightly browned.
Remove the muffin pan from the oven and immediately press a chilled peanut butter cup in the center of each cookie until it is level with the top of the cookie cup. Repeat with remaining peanut butter cups. Cool in the pan for 30 minutes. Remove from the pan and cool completely. To remove easily, run a small knife (like a steak knife) around the edge of each cookie cup to release it from the sides and it should lift out easily.