In a large bowl, beat the softened butter, granulated sugar, and brown sugar together until light and fluffy, about 2 minutes.
Add the egg and vanilla extract and mix until fully combined and smooth.
In a separate bowl, whisk together the all-purpose flour, cream of tartar, baking soda, and salt. Add the dry ingredients to the wet mixture and stir until a soft dough forms.
In a small bowl, mix together the granulated sugar and ground cinnamon for the coating.
Scoop tablespoon-sized balls of dough and roll each one generously in the cinnamon sugar coating until fully covered on all sides.
Place the coated dough balls in the air fryer basket in a single layer, leaving about 1 inch of space between each cookie. Work in batches as needed.
Air fry at 325°F (165°C) for 7-8 minutes, until the edges are set and the tops look slightly crinkled. The centers should still appear soft.
Let the cookies cool in the basket for 3 minutes before transferring to a wire rack. They will firm up as they cool.