Caramelize onions in olive oil and butter until golden and soft. Add garlic and cook briefly.

Reserve part of the onions. Mix remaining onions with ground beef, eggs, breadcrumbs, Worcestershire sauce, mustard, thyme, salt, and pepper.

Shape mixture into a loaf and bake until mostly cooked through.

Stir flour into reserved onions, then add broth and milk. Simmer until thickened.

Spoon onion sauce over meatloaf, top with cheeses, and bake until melted and golden. Rest before slicing.