Melt butter over flat top griddle or large cast-iron skillet and toast rye bread slices. Season and grill burgers on flat top. Cook to medium-rare. Put olive oil on griddle and grill onions until caramelized. Melt cheese slices over each burger. Assemble pattymelt by placing cheeseburgers over one slice each rye toast, top with caramelized onions and remaining slices of rye toast.
Serve with French fries and sliced cantaloupe.