Whisk the flour, baking soda, and salt together in a medium bowl. Set aside.
In a large bowl using a hand mixer or stand mixer fitted with the paddle attachment, cream the butter, peanut butter, brown sugar, and granulated sugar together on medium speed until combined. Scrape down the sides and bottom of the bowl as needed. On low speed, add the egg and vanilla extract. Turn up the mixer to high and beat until lighter in color and smooth— about 1-2 minutes. Scrape down the sides and bottom of the bowl as needed. On low speed, slowly add in the dry ingredients. Mix until combined.
Cover tightly and chill the cookie dough for at least 30 minutes or up to 3 days. Chilling is recommended for this recipe as the soft cookie dough can spread.
Preheat the oven to 350°F. Line two large baking sheets with parchment paper or silicone baking mats (preferred.) Set aside.
Pour the 1/3 cup granulated sugar in a small bowl for the topping. Grab the cookie dough from the refrigerator and measure cookies to be 1 heaping to 1.5 Tablespoons each. Roll each cookie dough ball in the granulated sugar and arrange at least 2 inches apart on the baking sheets.
Bake for 10 minutes. The cookies will look very soft— that's OK! Allow the cookies to cool on the baking sheet for 5 minutes. Then, quickly press 1 peanut butter cup into the center of each cookie. Transfer the cookies to a freezer-safe dish and freeze for 10-15 minutes or until the peanut butter cup has set. Some of the peanut butter cups may melt slightly and that's OK!
The cookies stay fresh covered at room temperature for up to 1 week.
Make ahead tip: The cookie dough can be made up to 4 days in advance and stored in the refrigerator covered. If chilling longer than 4 hours, remove the dough from the refrigerator at room temperature for 15-20 minutes prior to rolling. Baked cookies freeze well for up to 3 months. Thaw overnight in the refrigerator. Cookie dough balls can also be frozen for up to 3 months — no need to thaw, just bake an extra minute.Optional (but highly recommended): When oven(s) running, the house temperature can increase causing the Reese's to soften a bit. To prevent too much melting when placing it in the cookie, I refrigerate unwrapped Reese's until I'm ready to use them. For this recipe, I find using a non-natural peanut butter works best such as Jif, Skippy, etc.