Preheat oven to 375°F (190°C). Line baking sheets with parchment paper.

In a large bowl, cream together the softened butter, peanut butter, granulated sugar, and brown sugar until light and fluffy.

Beat in the egg and vanilla extract until well combined.

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

Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.

Roll dough into 1-inch balls and place them on the prepared baking sheets, about 2 inches apart.

Using a fork, press down on each cookie in a crisscross pattern to flatten slightly.

Bake for 9 to 11 minutes, or until the edges are lightly golden brown.

Let the cookies cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.