In a large skillet, heat the olive oil and butter over medium heat. Add the sliced onions and cook, stirring occasionally, until they are golden brown and caramelized, about 20-25 minutes. Stir in the balsamic vinegar and dried thyme, and cook for an additional 2 minutes. Remove from heat and set aside.
In a small bowl, combine the mayonnaise, Dijon mustard, and minced garlic. Mix well and set aside.
In a large mixing bowl, combine the ground chuck and the packet of dry French onion soup mix. Mix gently until just combined, being careful not to overwork the meat. Divide the mixture into four equal portions and shape them into patties.
Preheat your grill or grill pan over medium-high heat. Cook the burger patties for about 4-5 minutes on each side, or until they reach your desired level of doneness. During the last minute of cooking, top each patty with shredded gruyere cheese and close the grill lid to melt the cheese.
To assemble, spread a generous amount of the prepared sauce on the bottom half of each brioche bun. Add a handful of baby arugula, followed by the cheesy burger patty, and top with the caramelized onions. Place the top half of the bun on and serve immediately.