Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a large bowl, mix peanut butter, granulated sugar, brown sugar, egg, vanilla extract, baking soda, and salt until smooth.
Roll dough into small balls (about 1 tablespoon each) and place them 2 inches apart on the baking sheet.
Flatten each ball with a fork to create a crisscross pattern.
Bake for 8-10 minutes, or until edges are golden brown. The cookies will still be soft but will firm up as they cool.
Let the cookies cool completely before assembling the sandwiches.