1. Heat oil in a large cast iron skillet over medium heat. Add onions, garlic, and thyme to pan and cook, stirring occasionally, until onions begin to soften, 3–4 minutes.
2. While onions are cooking, season lamb with salt and pepper and form into 4 patties. Push softened onions to outer edge of skillet and add burgers. Cook until browned on one side, 2–3 minutes, then flip burgers over. Add ale and mustard and stir gently to combine. Continue cooking burgers to desired doneness, 4–6 minutes for medium. Transfer burgers to a plate and continue reducing liquid until it reaches a gravy-like consistency, about 10 minutes. Season with salt and pepper.
3. Spread mayonnaise on bottom half of each bun, top with lettuce, burgers, and onions with pan sauce.