In a stand mixer bowl, combine the brown sugar, white sugar, peanut butter, and butter. Beat these ingredients together at medium speed for about 3-4 minutes, or until the mixture is smooth and creamy.

Next, incorporate the egg and vanilla extract into the creamy mixture. Beat at medium speed for an additional 1-2 minutes until everything is fully combined.

In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.

Gradually add the dry ingredients to the wet mixture, mixing on low speed for about 2-3 minutes until the dough is evenly blended.

Transfer the cookie dough to a bowl, cover it with plastic wrap, and refrigerate for at least 1 hour.

Preheat your oven to 375°F (190°C).

Line a baking sheet with parchment paper or a silicone mat.

Roll about 1-½ tablespoons of chilled dough into balls and place them on the prepared baking sheet, leaving space between each ball. Press down with a fork to create a crisscross pattern.

Bake the cookies in the preheated oven for 9-11 minutes, until the edges are golden brown.

Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a cooling rack.