In a medium saucepan, melt the butter over medium heat. Continue to cook, swirling the pan occasionally, until the butter turns golden brown and has a nutty aroma, about 5-7 minutes. Remove from heat and let cool slightly.

In a large mixing bowl, combine the brown butter, peanut butter, granulated sugar, and brown sugar. Beat until creamy. Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.

In a separate bowl, whisk together the flour, baking soda, baking powder, and salt.

Gradually add the dry ingredients to the wet ingredients, mixing just until combined. The dough will be soft and slightly sticky.

Cover the dough and chill in the refrigerator for at least 30 minutes. This helps the cookies maintain their shape while baking.

Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.

Scoop out tablespoon-sized portions of dough and roll them into balls. Place the dough balls on the prepared baking sheet about 2 inches apart.

Press down slightly with a fork to create a crisscross pattern.

Bake for 10-12 minutes, or until the edges are lightly golden. The centers will still be soft but will firm up as the cookies cool.

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