Melt butter in skillet over medium-high heat. Add onions to hot pan and begin caramelizing, stirring occasionally. After 4 minutes or so, add apples to the onions and continue to cook until onions are golden brown, about 6 more minutes. Remove from heat when done.
While onions and apples are cooking, blend together in a large bowl the ground beef, salt, pepper, thyme, Dijon, egg, and bread crumbs. Be sure ingredients are well incorporated, but don't over work. Form mixture into 4 good-sized patties. (hint: If you create a concave burger, with the center thinner than the edges, when it cooks the burger won't swell up to look like a meatball, but will remain flat and patty-like)
Heat grill to medium-high heat.
Place burgers on grill and cook about 10 minutes per side, or until cooked through. As burgers are finishing (after the first flip), spread mayonnaise on both cut sides of each split roll. Place rolls on grill, mayonnaise side down, to toast for about 1 minute, or until golden.
Place each burger on the bottom half of a toasted roll. Top with about 1/4 cup of blue cheese, then with a generous scoop of the caramelized onion and apple mixture. Cover with the roll and serve.
