Break up any large lumps in 2 cups of cold mashed potatoes with a fork or potato masher so the mixture combines evenly.
Add the mashed potatoes, beaten egg, 3/4 cup shredded sharp cheddar, 2 tablespoons chives, 1/2 teaspoon garlic powder, 1/2 teaspoon onion powder, 1/2 teaspoon salt, and 1/4 teaspoon black pepper to a mixing bowl. Add 1/3 cup of flour and fold until just combined. If too sticky, add more flour 1 tablespoon at a time.
Line a baking sheet with parchment. Portion the mixture using a 1/4 cup measure or cookie scoop, then shape into 1/2-inch thick patties. Dampen your hands to prevent sticking. Freeze for 10 minutes for firmer cakes.
Preheat air fryer to 400°F for 3-5 minutes. Spray the basket generously with cooking spray.
Arrange patties in a single layer in the basket, leaving space between each. Lightly spray tops with cooking spray. Cook at 400°F for 8-10 minutes, flipping halfway.
At halfway, flip carefully using a thin metal spatula. Spray the top lightly again. Continue cooking until the second side is golden and crispy.
Serve immediately while hot and crispy with a dollop of sour cream, extra chives, bacon bits, or any desired toppings.