Line 2 large baking sheets with parchment paper. Set aside.
To the bowl of a stand mixer* fitted with the paddle attachment, add the butter, brown sugar, and peanut butter. Beat on medium-high speed until light and creamy (about 3-5 minutes).
Add the vanilla extract. Mix to combine.
Add the eggs one at a time, mixing well between each addition.
In a separate bowl, whisk together the flour, baking soda, cornstarch, and salt. Add the dry ingredients to the wet ingredients. Mix on low until just combined.
Fold in the peanut butter chips and Reese's pieces.
Chill the dough in the refrigerator for 15 minutes. Near the end of the chill time, preheat the oven to 400°F.
Using a ¼ cup cookie scoop or measuring cup, scoop the dough out onto the prepared baking sheets, about 3 inches apart. Using your hands or the bottom of a glass, slightly flatten each ball of dough. (These cookies don't spread.)
Bake for 8-10 minutes, or until the edges are set and the tops are golden brown. The center of these cookies will be gooey, but they will continue to bake out of the oven.
Allow the cookies to cool on the baking sheet for at least 30 minutes before serving.
Notes
*If you do not have a stand mixer you can use a hand-held mixer.