Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper. Set aside.
In a large mixing bowl, use a hand mixer or stand mixer to cream together the peanut butter, unsalted butter, granulated sugar, and brown sugar until smooth and fluffy (about 2 minutes).
Add the egg and vanilla extract, mixing until fully incorporated.
In a separate bowl, whisk together the flour, baking powder, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until a soft dough forms.
Using a 1-tablespoon scoop, drop cookie dough onto the lined baking sheet about 2 inches apart. Use a fork to gently press each dough ball, creating a crisscross pattern. This helps flatten the cookies and gives them the classic peanut butter cookie look.
Bake for 10-12 minutes, or until the edges are golden and the tops look set. Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.