Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking soda, and salt. Set aside.
In a large bowl, use an electric mixer to cream together the softened unsalted butter, granulated sugar, and light brown sugar until light and fluffy, about 2-3 minutes.
Add the creamy peanut butter to the butter and sugar mixture and beat until combined.
Beat in the egg and vanilla extract until fully incorporated.
Gradually add the flour mixture to the wet ingredients, mixing on low speed until just combined. Do not overmix.
Roll the dough into 1-inch balls and place them about 2 inches apart on the prepared baking sheets.
Use a fork to gently press down on each ball in a crisscross pattern to flatten slightly.
Bake in the preheated oven for 9-11 minutes, or until the edges are lightly golden but the centers are still soft.
Remove the cookies from the oven and allow them to cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.