Preheat the oven to 350℉. Spray two 12-cup muffin tins with nonstick spray.
In a medium bowl, whisk together the flour, baking soda, baking powder and salt.
In a large bowl, use an electric mixer to combine the peanut butter and butter. Add the sugars and mix until combined. Beat in the egg. Add in the dry ingredients a little at a time and mix just until they are incorporated.
Fill 16 muffin cups about half-full with cookie dough. Bake for 14 to 16 minutes, or until the cookie cupcakes are golden on the edges. Expect them to sink slightly in the centers. When they come out of the oven, place a peanut butter cup in the center of each warm cookie cupcake and press down lightly. Place the cupcake pans in to the refrigerator or freezer to expedite the cooling process, if desired. Once the peanut butter cups have hardened, remove the cookie cupcakes from the pan.