Note; You can make this recipe on a grill using a griddle or indoors, using a skillet (preferably cast iron).
Heat the oil on a grill griddle or in a large skillet over medium-high heat.
Toss the onions with the rosemary and 1/4 teaspoon of salt. Transfer to the griddle or skillet and cook 8 minutes or until starting to soften. Turn and cook another 8 minutes or until the onions are completely soft and caramelized. Add more oil as the onion cooks if they are too dry but do not drench them. Remove to a bowl, cover and keep warm until ready to use.
Shape the beef into 4 equally-sized patties. Season well on both sides with remaining salt and all of the pepper.
Transfer the patties to the grill or skillet and cook 5-6 minutes per side until done as desired. I prefer thin patties, so I smash my patties onto my griddle as they start to cook. Just before removing the patties, top each with some of the caramelized onion and the blue cheese crumbles.
Toast or grilled the buns as desired.
Top the buns with the patties. Add lettuce and tomato and serve.