In a medium bowl, combine the peanut butter, granulated sugar, and brown sugar. Stir vigorously with a wooden spoon or spatula until completely smooth and well blended.
Add the egg, vanilla extract, baking soda, and salt. Mix until fully incorporated. Then add the flour and stir just until a soft, scoopable dough forms. Do not overmix.
Scoop rounded tablespoon-sized portions of dough and roll them into balls. Press each ball gently with the tines of a fork in a crosshatch pattern to flatten slightly to about 1/2 inch thick.
Preheat the air fryer to 325 degrees F for 2 minutes. Cut a piece of parchment paper to fit the air fryer basket.
Place the cookies on the parchment in the air fryer basket, leaving about 1 inch of space between each one. Cook in batches as needed.
Air fry at 325 degrees F for 7 to 8 minutes, until the edges look set but the centers still appear slightly soft. They will firm up as they cool.
Let the cookies cool in the basket for 3 minutes before transferring them to a wire rack. They will be soft and chewy in the center with lightly crisp edges once fully cooled.