To make the onions, melt butter with oil in a large pan over medium high heat. Add sliced onion and season with salt.
Reduce heat to medium-low and cook for 45 to 60 minutes, stirring occasionally, until the onions are a deep brown. In the last 10 minutes of cooking, deglaze the pan with balsamic vinegar, stirring to combine. Remove from heat and set aside.
In a medium bowl, combine ground beef, garlic powder, onion powder, thyme, salt, pepper, and Worcestershire sauce. Mix together with your hands until well combined. For four equal sized patties.
Cook patties for 2 to 3 minutes per side on a medium-high grill or in a cast iron skillet over medium-high heat, or until your desired doneness. In the last minute of cooking, top with equal amounts of shredded cheese and cover until cheese is melted.
To assemble, add a piece of butter lettuce to the bottom halves of the buns. Top with burgers, then caramelized onions. Place the top halves of the buns on top and serve.