Place the meat in a bowl and add the chopped garlic, gently incorporating. Form into four patties and drizzle olive oil on each side. Season each side with a sprinkle of salt and set aside until ready to grill. You can form these a few hours ahead of time and place in the refrigerator.
Heat the oil in a saute pan and cook the onions and teaspoon of salt over medium heat for 15 minutes until softened, stirring occasionally. Add the mushrooms and cook until the liquid from the mushrooms has cooked off, about another 15 minutes. Meanwhile, pour the barbecue sauce in a small pan with plenty of freshly cracked black pepper and cook until reduced and thickened considerably. It should be the consistency of thick ketchup. The amount of time it takes will depend on the brand of sauce. It could take up to 20 minutes at a low simmer. Taste and season with salt and pepper if needed. Keep the sauce warm while you make the burgers.*
Heat your grill, or warm a saute pan, and cook the patties, turning once or twice, until cooked through for turkey, or until your desired doneness for beef. The amount of time will depend on the meat that you use, and how you like your burger.
Serve the burgers with the buns, sauce, mushrooms and onions, and top with the blue cheese crumbles.