In a large saucepan, heat oil over medium heat. Add onions and garlic and cook until soft. Add tomato sauce, ketchup, mustard, brown sugar, apple cider vinegar, Worcestershire sauce, chili powder and sea salt. Mix to combine and cook for another minute. Stir in turkey. Stir in cheese until melted. Remove from heat.
Slice buns and toast insides on a hot pan. Add a layer of lettuce. Spoon in turkey mix. Serve immediately.