In a large bowl, beat the peanut butter and softened butter together until smooth and creamy, about 1 minute.
Add the granulated sugar and brown sugar and beat until light and fluffy, about 2 minutes. Mix in the egg and vanilla extract until well combined.
In a separate bowl, whisk together the all-purpose flour, baking soda, and salt. Add the dry ingredients to the wet mixture and stir until just combined into a soft dough.
Scoop tablespoon-sized balls of dough and roll them between your hands. Press each ball gently with the back of a fork in a criss-cross pattern to flatten slightly.
Place the cookies in the air fryer basket in a single layer, leaving about 1 inch of space between each cookie. You will need to work in batches.
Air fry at 325°F (165°C) for 7-8 minutes, until the edges are set but the centers still look slightly soft and underdone.
Let the cookies cool in the basket for 3 minutes before carefully transferring them to a wire rack to cool completely. They will firm up as they cool.