Let 2 tablespoons unsalted butter sit at room temperature until softened. Heat an air fryer to 375ºF.
Grate 4 ounces sharp cheddar cheese and 4 ounces Monterey Jack cheese on the large holes of a box grater (about 1 cup each). Toss together in a medium bowl.
Spread the butter onto one side of each of 4 slices sandwich or rustic bread. Flip 2 of the slices over and divide the cheese evenly over these slices. Close the sandwiches with the remaining 2 bread slices butter-side up. Press the sandwiches slightly to close.
Air fry the grilled cheeses: place in the basket (do one at a time if they will not fit in a single layer with some space around them) and air fry for 3 minutes. Flip the sandwich and air fry until golden brown, 3 to 4 minutes more.