For the onions: Heat olive oil in a large skillet over medium-high heat. Add onions and saute until they begin to brown, 4 to 5 minutes. Add beef broth and use a wooden spoon to release any browned bits off the bottom of the skillet. Reduce heat to low and let cook, stirring occasionally, until the onions are caramelized, about 20 minutes. Stir in vinegar and season to taste with salt and pepper. Remove from heat.
For the burgers: Combine beef, onion and soup mix in a mixing bowl. Shape into 4 equal patties.
Preheat a griddle over medium-high heat. Cook burgers 5 to 7 minutes, then flip, top with Gruyere and cover. Let cook until cheese melts and burgers reach desired doneness, 4 to 6 minutes more. Serve on buns with caramelized onions and arugula.