In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
In a stand mixer fitted with the paddle attachment, cream together the butter, peanut butter, and brown and granulated sugars on medium speed until light and fluffy, about 2 minutes. Scrape down the sides of the bowl with a rubber spatula. Mix in the egg until well incorporated. Mix in the vanilla. Add the flour mixture all at once and mix on low speed just until it all comes together. Scrape down the bowl, turn the speed to medium, and mix for 15 seconds.
Divide the dough into two disks, wrap them in plastic, and refrigerate until firm, at least an hour. The dough can also be made a couple of days ahead or frozen for up to 3 weeks.
Preheat the oven to 350°F / 175°C. Line baking sheets with parchment paper.
Roll one of the disks of dough out between two pieces of parchment paper to a rectangle 1/8 inch / 3mm thick.
The dough may crack slightly along the edges. You can work them back together with your hands, but don’t fret too much, because the results will still be amazing.
Use a cookie cutter to create fun shapes out of the dough. You can gather the scraps, refrigerate, and reroll once before the dough gets too tough. Place the cut cookies on the prepared baking sheet, leaving about 3/4 inch of space between them. If your dough starts to get warm and sticky, return it to the refrigerator. Warm dough will be difficult to work with.
Bake, one sheet at a time, in the middle of the oven for 8 to 12 minutes, until the bottoms turn light golden brown. The baking time will vary based on the size of the cookie.
Allow to cool completely before filling. The cookies can be made a few days before and stored in an airtight container or wrapped and frozen for up to 1 month.