Combine butter, Gouda cheese, Cheddar cheese, sea salt, black pepper, and cayenne pepper in a food processor; pulse until butter is in bits and the mixture forms small curds. Add flour; pulse until dough looks moist and forms large popcorn-sized curds.

Turn dough out onto a flat surface; knead gently just until it comes together and you can shape it into a ball. Divide into 2 pieces. Pat each piece into a disk.

Place 1 disk between 2 sheets of parchment paper. Roll to a thickness of 1/4 inch. Repeat with second disk. Stack sheets of dough on a baking sheet. Freeze until firm, about 1 hour.

Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper or a silicone baking mat.

Peel parchment paper off 1 sheet of frozen dough. Cut into cookies using a 1 1/2-inch-diameter cutter; arrange 1 inch apart on the prepared baking sheet. Repeat with the second sheet of dough.

Bake cookies in the preheated oven until lightly golden on the bottom, about 9 minutes. Rotate baking sheet. Continue baking until bottoms are golden brown and tops are lightly golden, about 5 minutes more. Cool on the baking sheet for 2 minutes. Transfer to a rack to cool completely, about 10 minutes.

Gather dough scraps, roll to a thickness of 1/4-inch, and freeze until firm, about 15 minutes. Cut into cookies and bake.