In a large bowl, cream together butter, granulated sugar, and brown sugar until light and fluffy using an electric mixer, then beat in the egg and vanilla extract; finally, add peanut butter and mix until incorporated, scraping the bowl.

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

Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined, being careful not to overmix.

Cover the dough, chill it in the refrigerator for at least 30 minutes, preheat your oven to 375°F (190°C), and line baking sheets with parchment paper.

Roll the dough into 1-inch balls, place them 2 inches apart on the prepared baking sheets, and use a fork to gently press a crisscross pattern onto the top of each cookie, flattening them slightly.

Bake for 8-10 minutes until the edges are lightly golden and the centers are still slightly soft, then remove from the oven and cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.