For the topping:
For the burgers:
To make the topping, place a large skillet over medium heat. Cook the bacon until lightly browned but not yet crisp, about 8 minutes. Transfer the bacon to paper towels to drain. Drain all but 2-3 tablespoons of the bacon grease from the skillet. Add the onion to the skillet, season with salt and pepper, cover the pan and cook for two minutes. Uncover the pan, add a splash of water and scrape any browned bits from the bottom of the pan. Cover and cook, stirring occasionally, until the onions are soft and beginning to brown, about 10 minutes.
Stir in the vinegar, mustard and water. Return the bacon to the pan and bring the mixture to a simmer. Simmer uncovered until the mixture has thickened and most of the liquid has been absorbed, 2-4 minutes. Transfer to a small bowl and let cool slightly. Cover and refrigerate if making in advance.
To make the burgers, finely mince the bacon. Add to a bowl with the ground beef, Worcestershire, salt and pepper. Mix well with a fork until evenly combined. Divide into four equal portions and form into patties about ¾-1 inch thick.
Heat a grill to medium-high heat. Grill to desired doneness, about 4-5 minutes per side depending on preference and grill temperature. When the burgers are almost finished cooking, top with the Gruyere, if using. Remove the patties to a plate and let rest. Meanwhile briefly toast the buns on the grill. Assemble the burgers on the toasted buns and topped with the onion jam.