Directions

1. Heat butter and oil in a large skillet over medium until melted. Stir in onions, thyme, 1 teaspoon salt, and 1/2 teaspoon pepper. Cook, stirring occasionally, until onions are translucent, about 3 minutes. Reduce heat to medium-low; cover and cook, stirring occasionally, until onions begin to soften and brown around edges, about 18 minutes. Uncover and increase heat to medium-high; stir in 1/4 cup sherry, scraping up any browned bits from bottom of skillet. Cook, stirring often, until sherry is evaporated and onions are browned and caramelized, 3 to 4 minutes.
2. Divide meat into 4 (6-ounce) portions. Working with 1 portion a time, gently shape into a ball. Flatten ball between the palms of your hands, and shape into a 4 1/2-inch-wide patty (about 3/4-inch thick). Press your thumbs into center of patty, and create an indention (about 1/2-inch deep).
3. Preheat a 6-quart air fryer to 400°F for 5 minutes. Sprinkle patties evenly with remaining 1 teaspoon salt and 1 teaspoon pepper. Working in batches, place 2 patties in a single layer in preheated air fryer basket; cook until patties form a browned crust on edges and a meat thermometer inserted into center of patties registers 130°F for medium-rare, 5 minutes, or until desired degree of doneness. Remove cooked patties; set aside. Repeat cooking process with remaining 2 patties. When remaining patties are cooked, transfer all patties back to air fryer. Place 1 cheese slice on each patty, and close air fryer; let stand just until cheese is melted, about 30 seconds. Remove patties; let rest for 2 to 3 minutes.
4. While patties are cooking, whisk together mayonnaise, Worcestershire sauce, mustard, soy sauce, and remaining 1 teaspoon sherry in a small bowl. Set aside.
5. Spread 1 tablespoon mayonnaise mixture on each of the top and bottom bun halves. Place 1 meat patty on each of the bottom bun halves. Divide caramelized onions into 4 portions (about 1/4 cup each), and place 1 portion on top of each meat patty; add top bun halves. Serve immediately.