Prepare the Burger
Cut brioche buns in half and brush each cut side with melted butter. Set aside.
Divide ground beef into four equal portions (eight ounces each). Shape patties into uniform, but loosely packed burgers about an inch larger in diameter than the buns. Make a depression in the center of each patty. Right before placing on heat, season burgers on both sides generously with Kosher salt and fresh cracked black pepper.
ON A GRILL: Turn gas grill to high or bring a charcoal to high heat (minimum of 600 degrees). Cook burgers on one side, then flip and continue cooking. Allow 3 minutes per side for medium rare or 4 minutes for medium. Top with Gruyere, close lid and allow to cook about one more minute or until the cheese is just melted. Remove burgers and place on platter.
PREPARE IN A FRY PAN OR GRIDDLE: Heat a tablespoon of oil in the pan until hot. Place patties on hot surface and cook until brown and a nice crust forms, about 3 minutes. Flip the burgers and cook another 3 minutes for medium rare or 4 for medium. Top with Gruyere slices and cover for one more minute to let cheese melt. Remove burgers and place on platter.
Place buns on grill or pan to toast. Remove and immediately build the burgers. Spread about 1 1/2 tablespoons aioil on bottom bun. Cover with a layer of arugula. Add patty and cheese. Place about 2 tablespoons onion jam on top and cover with the toasted bun top. Serve and enjoy!
Rosemary Aioli
Mix mayonnaise, rosemary, garlic, lemon juice, salt, and pepper in a bowl. Cover and refrigerate for an hour or more before serving.
Red Onion Jam
Heat canola oil in a skillet over medium heat. Add onions, season with salt and pepper, and cook until softened, about 5 to 8 minutes.
Add wine, sugar and balsamic. Bring the contents to a boil, then reduce the heat to low. Simmer until liquid thickens, stirring occasionally. Taste and add additional salt and pepper if needed. Transfer finished jam to a bowl to cool. Use right away or store in the fridge until ready to use. Keeps for about 10 days.