Line a buttered 7-inch metal cake pan with parchment paper or a silicone liner.
Combine the flour and sugar in a large mixing bowl. Grate the butter into the flour. Add the grated cheese and stir everything to coat the cheese and butter with flour. Then add the buttermilk and stir it just until you can no longer see streaks of flour. The dough should be quite wet.
Spread the all-purpose (not self-rising) flour out on a small cookie sheet. With a spoon, scoop 8 evenly sized balls of dough into the flour, making sure they don’t touch each other. With floured hands, coat each dough ball with flour and toss them gently from hand to hand to shake off any excess flour. Place each floured dough ball into the prepared pan, right up next to the other. This will help the biscuits rise up, rather than spreading out.
Pre-heat the air fryer to 380F.
Transfer the cake pan to the basket of the air fryer, lowering it into the basket using a sling made of aluminum foil (fold a piece of aluminum foil into a strip about 2-inches wide by 24-inches long). Let the ends of the aluminum foil sling hang across the cake pan before returning the basket to the air fryer.
Air-fry for 20 minutes. Check the biscuits a couple of times to make sure they are not getting too brown on top. If they are, re-arrange the aluminum foil strips to cover any brown parts.
After 20 minutes, check the biscuits by inserting a toothpick into the center of the biscuits. It should come out clean. If it needs a little more time, continue to air-fry for a couple extra minutes. Brush the tops of the biscuits with some melted butter and sprinkle a little more grated cheese on top if desired.
Pop the basket back into the air fryer for another 2 minutes. Remove the cake pan from the air fryer using the aluminum sling. Let the biscuits cool for just a minute or two and then turn them out onto a plate and pull apart. Serve immediately.