Preheat the air fryer to 370°F (187°C). Place the patties in the air fryer basket, ensuring they're not touching. Cook for about 8-10 minutes, flipping halfway through, or until they reach your desired doneness. Remove the patties and set aside.

While the patties cook, caramelize the onions in a skillet over medium heat with a little butter, stirring occasionally until they're golden brown and soft. This could take about 15-20 minutes for the best flavor.

Butter one side of each slice of rye bread. On the unbuttered side, layer a slice of Swiss cheese, a cooked beef patty, a generous helping of caramelized onions, another slice of cheese, and finally, top with another slice of bread, buttered side up.

Place the assembled sandwiches back in the air fryer. You might need to cook them one at a time, depending on the size of your air fryer. Cook at 370°F (187°C) for about 3-4 minutes per side, or until the bread is toasted to your liking and the cheese has melted.

Once cooked, remove the patty melts from the air fryer. Let them sit for a minute before cutting into halves. Serve hot for the best experience.