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

In a large mixing bowl, combine peanut butter, softened butter, granulated sugar, and brown sugar. Beat with a mixer on medium speed until smooth and fluffy.

Add the egg and vanilla extract to the creamed mixture, mixing until well integrated.

In a separate bowl, sift together flour, baking soda, and salt. Gradually pour this dry mixture into the wet ingredients, mixing on low speed until a smooth dough forms.

Roll portions of dough into 1-inch balls and place them on the prepared baking sheet, flattening each ball with a fork to form a crisscross pattern.

Bake cookies for about 10 minutes until lightly golden around the edges but soft in the center.

Let cookies cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.