In a small bowl, whisk together the 1/4 cup Thousand Island dressing, 1 tbsp ketchup, 1 tsp Worcestershire sauce and 1 tsp yellow mustard until smooth. Set aside.
Form the 1 lb ground beef into 4 thin patties (slightly larger than your bread slices, as they’ll shrink during cooking). Season with Salt and pepper to taste on both sides.
Preheat the air fryer to 375°F (190°C) for about 5 minutes.
Place the beef patties in the air fryer basket in a single layer. Air fry for 8-10 minutes, flipping halfway through, until fully cooked and slightly crispy on the outside (internal temperature should reach 160°F or 71°C). Once done, place a slice of 4 slices American cheese on two of the patties and a 4 slices Swiss cheese on the other two. Let the cheese melt while the patties rest.
2 tbsp butter one side of each 8 slices sourdough bread. Place 2 slices, buttered side down, in the air fryer basket. Spread a layer of Frisco sauce on the non-buttered side, and then add one Swiss-cheese patty and one American-cheese patty. Top with another slice of bread, buttered side facing up. Air fry the sandwiches for 3-4 minutes at 370°F, until the bread is golden brown and crispy, then flip and cook for another 2-3 minutes to toast the other side.