Preheat the oven to 355°F (180°C) and line two baking sheets with parchment paper.
In a bowl, whisk together the flour, baking soda, and salt until well combined.
In a stand mixer fitted with a paddle attachment (or use a hand mixer and a bowl or just a bowl and a whisk), cream together the butter and peanut butter until fluffy.
Add the white and brown sugar and beat for 2 minutes.
Add the egg and vanilla, and beat until well combined.
Add the dry ingredients mixture and mix until just combined.
Form 16 dough balls (approximately 35g each), and place them on the prepared baking sheets, spacing them about 2 inches apart.
Bake the cookies in two batches for 11-12 minutes or until the edges are lightly golden, the cookies are puffy, and they have finished spreading.
Remove the cookies from the oven and drop the baking sheet a couple of times on a flat surface to deflate the cookies. Let them cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.