In a small bowl, whisk together the flour, baking soda and salt; set aside.
Beat together the butter, peanut butter and both sugars on medium speed until light and fluffy, about 3 minutes, scraping the sides of the bowl as needed. Add the egg and vanilla and mix for another minute or so to combine. Reduce the mixer speed to low, add the dry ingredients, and mix until just combined. Give the dough a final mix with a rubber spatula to ensure all of the flour is incorporated.
Shape the dough into 1-inch balls (about 2 teaspoons worth of dough) and place in the cups of a miniature muffin pan (keep the dough in balls - do not press into the pan).
Bake for 8 to 10 minutes, or until light golden brown and slightly puffy. Remove from the oven and immediately press a miniature peanut butter cup in the middle of each cookie, pressing down so that the top of the peanut butter cup is even with the top of the cookie. Cool for at least 10 minutes before removing from the pans, and let cool completely. Store in an airtight container at room temperature for up to 1 week.
Use a traditional peanut butter such as Jif or Skippy; natural-style peanut butters will be too oily for this recipe.
If you don't have a mini muffin pan, you can make these on a traditional cookie sheet, however they will spread out when you press the peanut butter cup in the middle, so you will have less of a tassie-style cookie and more of a regular-size cookie.
Don't overbake the cookies! You want them to look puffy and pale when you pull them from the oven; if they get golden brown, the peanut butter cup won't easily sink down into the cookie.
Store the peanut butter cup cookies in an airtight container at room temperature for up to 1 week, or freeze for up to 3 months.