Melt butter with olive oil in a thick-bottomed pan until the oil is shimmering. Add the sliced onions and stir to coat. Cook, stirring on occasion for 7 to 10 minutes. Do not let the onions burn or dry out. Sprinkle the vinegar, salt and sugar over the onions and continue cooking for about 25 to 30 minutes, stirring frequently. Continue to cook and stir, until soft and translucent.
Preheat gas or charcoal grill to medium-high heat.
In a large bowl, combine beef, Worcestershire sauce, mustard, garlic, paprika. salt and pepper. Use your hands to mix and knead the mixture. Divide into 4 even portions and shape into patties.
Lightly oil grill grate.
Place the burgers on the hot grill grate. Grill the burgers until cooked through, about 5 to 7 minutes per side for medium. To check for doneness, insert an instant-read thermometer through the side of the burger until you reach the center. The internal temperature should be at least 160º F.
Brush the buns with melted butter and toast on the grill for 30 to 45 seconds, or until golden brown.
Assemble the burgers: Place greens on the bottom bun, season to taste with salt and pepper. Arrange the burger and goat cheese on top of the lettuce leaves. Top with caramelized onions. Add your favorite condiment, and remaining bun top. Serve.