Preheat the oven to 350°F and position a rack in the center. Lightly coat the wells of a nonstick mini muffin pan with nonstick spray.

In a small bowl, whisk together the flour, salt, baking soda, and baking powder, then set aside.

In the bowl of a stand mixer fitted with a paddle attachment, or in a large bowl using a hand mixer, combine the peanut butter and butter. Beat on medium speed until smooth and creamy, about 30 seconds to 1 minute, scraping the bowl as necessary.

Add both sugars to the mixture and continue beating until thick and fluffy, about another 30 seconds to 1 minute. Add the egg and vanilla extract, then beat until the mixture becomes lightened and fully blended, about 15 seconds.

With the mixer on low speed, gradually incorporate the dry ingredients. Mix just until no visible flour remains, then give the dough one final stir by hand with a spatula to ensure complete mixing.

Using a 1 1/2-inch scoop or a tablespoon, portion the dough into heaping tablespoons. Roll each portion into a ball and place into the cavities of the prepared muffin tin. Do not flatten.

Bake for 8 to 12 minutes, rotating the pan halfway through. The cookies should appear puffed, with matte tops and edges just beginning to brown.

While cookies bake, unwrap the mini peanut butter cups.

Once baked, transfer the pan to a wire rack. Immediately press a mini peanut butter cup into the center of each hot cookie. Allow to cool in the pan for about 5 minutes until slightly set.

Carefully twist each cookie to release from the pan and transfer to a cooling rack. If needed, let the cookies rest a few more minutes in the pan before transferring.

Serve once completely cooled or still slightly warm with melty centers.