Make the bacon onion jam: Cook bacon in a skillet until crisp. Remove the bacon, leaving about 1 tablespoon of bacon fat behind.
Add butter and sliced onion. Cook over medium-low heat for about 15–20 minutes, stirring occasionally, until deeply softened and caramelized.
Add garlic and cook for 1 minute. Stir in balsamic vinegar and bacon and cook another 2–3 minutes, until thick and jammy. Season with salt and pepper.
Cook the beef: Brown ground beef in a skillet over medium-high heat, breaking it into crumbles. Season generously with salt and pepper and drain excess grease.
Preheat oven to 350°F. Slice the Hawaiian rolls horizontally and place the bottom half on a parchment-lined baking sheet or baking dish.
Spread mayonnaise over the rolls. Layer with the ground beef, bacon onion jam, Gruyère and blue cheese.
Add the top half of the rolls and cover with foil. Bake for 15 minutes, then uncover
Remove the tops briefly, add a generous handful of fresh arugula, replace the tops, then slice and serve.