In a large mixing bowl or stand mixer, cream together the brown sugar, white sugar, peanut butter, and softened butter until smooth and fluffy.

Beat in the large egg and vanilla extract until well combined.

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

Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined.

Cover the cookie dough and refrigerate for about 1 hour.

Preheat the oven to 375°F (190°C).

Roll the dough into balls and place on a lined baking sheet, flattening with a fork in a crisscross pattern.

Bake for 9-11 minutes, until the edges are golden brown.

Let cool on the baking sheet for about 5 minutes, then transfer to a wire rack.