01 - Heat butter and olive oil in a large skillet over medium-low heat. Add sliced onions and salt. Cook, stirring occasionally, for 20-25 minutes until onions are deeply golden and soft. Stir in sugar and balsamic vinegar for extra flavor. Remove from heat and set aside.
02 - In a bowl, gently mix ground beef with Worcestershire sauce, salt, and pepper. Do not overwork the meat. Divide into 4 equal portions and shape into patties slightly larger than the buns to account for shrinkage.
03 - Heat a grill or skillet over medium-high heat. Cook patties for 3-4 minutes per side for medium doneness, or adjust timing to preferred doneness. Top with cheese slices in the last minute of cooking, covering to melt.
04 - Lightly toast the burger buns on the grill or in a pan until golden brown and slightly crispy.
05 - Place lettuce and tomato on the bottom bun, add the burger patty, top generously with caramelized onions, add pickles and condiments as desired. Finish with the top bun and serve immediately.