Preheat oven to 350ºF (175ºC). In the large bowl of a standing mixer, beat butter, peanut butter, and sugars together until light and fluffy. Add egg and vanilla extract, beating to combine.
In a separate bowl, whisk together flour, baking soda, and salt. With mixer speed on low, gradually add to peanut butter mixture, beating to combine. Stir in chopped peanut butter cups.
To form cookies, roll a heaping tablespoonful of dough into a ball between your hands and flatten it with your palm onto an ungreased cookie sheet; space cookies about an inch apart. Bake 9-11 minutes, until edges start to turn golden brown but centres are still soft.
Cool on wire racks.