Line two baking sheets with silicone baking mats or parchment paper. Set aside. Chill peanut butter cups in the refrigerator.

Place the butter, peanut butter, brown sugar, and granulated sugar in a medium mixing bowl. Use a mixer to beat on medium speed until well combined, about 1 minute.

Add the egg and vanilla extract and continue to beat until smooth, about 30 seconds.

Add the flour, cornstarch, baking soda, and salt to the bowl. Beat on low speed, scraping the sides of the bowl with a rubber spatula as needed, until a smooth dough forms.

Remove peanut butter cups from refrigerator.

Divide the cookie dough into 12 balls, using roughly 3 tablespoons of dough for each. Gently press a ball into a 1/2-inch-thick disc in the palm of your hand. Place a peanut butter cup in the center of the disc, and enclose the dough around it. Place the dough ball on a small baking sheet or large plate. Repeat with the remaining cookie dough.

Transfer the dough balls to the freezer for 10-15 minutes.

Preheat the oven to 350°F.

Remove the dough balls from the freezer and arrange on the prepared baking sheets, leaving 2-3 inches between the cookies. Bake the first baking sheet of cookies on middle rack of oven until puffy and golden brown at the edges, 15-16 minutes.

Check to see that cookies are done. Remove from oven or add time as needed. Repeat the baking process with the second baking sheet.

Allow the cookies to rest on the sheets for 5-10 minutes, then transfer to a wire rack to cool completely before serving.