Use a mandoline or sharp knife to slice the potatoes into thin, even rounds about 1/8-inch thick. Place them in a bowl of cold water to prevent browning while you prepare the sauce.
In a medium saucepan over medium heat, combine the heavy cream, whole milk, minced garlic, flour, dried thyme, salt, black pepper, and nutmeg. Whisk continuously and bring to a gentle simmer. Cook for 2 to 3 minutes until the mixture thickens slightly, then remove from heat and stir in 1 cup of the shredded cheddar cheese until melted and smooth.
Grease a 6 or 7-inch round baking pan (one that fits your air fryer) with the butter. Drain and pat the potato slices dry thoroughly.
Layer about a third of the potato slices in the pan, overlapping them slightly. Pour a third of the cream sauce over the top. Repeat with two more layers of potatoes and sauce, finishing with sauce on top.
Cover the pan tightly with aluminum foil. Preheat the air fryer to 320 degrees F and place the covered pan inside. Cook for 30 minutes.
Remove the foil and sprinkle the remaining cheddar cheese and gruyere cheese over the top. Increase the air fryer temperature to 360 degrees F and cook uncovered for another 10 to 12 minutes until the cheese is bubbly and golden brown and the potatoes are tender when pierced with a knife.
Let the scalloped potatoes rest for 5 minutes before garnishing with fresh chopped chives. Serve warm.