In a large bowl, combine red onion, garlic, Worcestershire sauce, and pepper. Add ground beef to the mixture and gently mix with your hands just until combined, being careful not to overmix.

Form the ground beef mixture into evenly sized patties about ¾-inch thick.

Cook the patties.

Directions for grilling: Place the patties on the grill over medium heat, sprinkle with salt, and close the lid. Grill the patties 5 to 6 minutes on each side until the meat's internal temperature reaches 160°F. Once patties are to the proper temperature, add the blue cheese on top of each patty while the patties are still on the grill, cover with the lid, and grill for an additional minute allowing the cheese to melt. Remove patties from the grill.

Directions for broiling: Preheat the oven to broil on high. Place patties, evenly spaced, on a broiler pan and sprinkle with salt. Broil 4 inches from heat until the meat's internal thermometer reaches 160°F, about 4–6 minutes on each side. Add blue cheese on top of each patty and continuing broiling just until cheese is melted. Remove patties from the oven.

Directions for the stovetop: Sprinkle the patties with salt. In a lightly greased skillet, cook patties over medium heat until the meat's internal thermometer reaches 160°F, 5-7 minutes on each side. Add blue cheese on top of each patty and continuing cooking just until cheese is melted. Remove patties from the heat.

Preheat oven to 425°F.

Cut off the tops from the heads of each whole garlic to expose the cloves. Place on a piece of aluminum foil and drizzle lightly with olive oil. Wrap up the foil into a tight pouch, place in the oven, and roast for 45 minutes. Remove from the oven and allow the garlic to cool. Press the bottom of the garlic cloves to remove the garlic pulp from the skin. 

In a food processor, combine the roasted garlic, olive oil, lemon juice, mayonnaise, dijon mustard, cayenne, salt, and pepper. Pulse until smooth and well combined.

Add olive oil to a large cast iron or heavy-bottomed skillet over medium heat. 

Add the onions to the skillet, stirring to coat with the oil. Spread the onions out evenly in the skillet and let cook, stirring occasionally, until the onions become soft and translucent, and turn a beautiful caramelized blonde color, about 15-20 minutes. If the onions look like they are browning too quickly, add a little splash of water to the skillet and scrape the caramelized fond from the bottom of the pan for the onions to absorb.

Add salt and a generous drizzle of balsamic vinegar to the onions. Turn the heat to medium-low and continue to stir until the onions have developed a deep, rich browned color, about 10 minutes more. Once the onions have caramelized, remove skillet from heat.

Slather the bottom burger bun with roasted garlic aioli and top with the blue cheese burger patty, caramelized onions, tomato slices, and arugula. Give a good schmear of bacon jam to the remaining burger bun and place on top of your burger masterpiece.

Eat and be so very happy.