Heat oil in a 10-inch skillet over medium low heat. Add onion; season with salt and pepper to taste. Saute for about 30 minutes or until lightly browned; stir occasionally.
Meanwhile, mix together mayo, red wine vinegar, Garlic Galore Seasoning, and blue cheese. Set aside.
Mix together ground beef and Java Rub. Shape into 4 patties. Heat grill or grill pan over medium high heat. Grill burgers to desired doneness.
Toast/grill buns if desired.
Assemble burgers with arugula, hamburgers, blue cheese sauce, and caramelized onions.
