Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a large bowl, mix together the peanut butter, granulated sugar, and light brown sugar until well combined and creamy.
Add the egg and vanilla extract to the peanut butter mixture and beat until smooth.
In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined and a dough forms.
Roll the dough into 1-inch balls and place them about 2 inches apart on the prepared baking sheets.
Using a fork, gently press down on each ball to create a crisscross pattern.
Bake in the preheated oven for 9 to 11 minutes, or until the edges are set and lightly golden but the centers remain soft.
Remove the cookies from the oven and allow them to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.