Caramelize onions: In a large pan over medium heat, heat oil with butter. Add onion and cook until onion begins to sweat and turn golden, about 5 minutes. Reduce heat to medium-low and continue to cook until onions are soft, about 15 minutes more.
Add a splash of water to pan, and scrape up any browned bits. Continue to cook until onions are jammy and caramel in color, adding a splash of water if onions start to dry out, about 10 minutes more.
Season with salt and pepper, then stir in vinegar.
In a small bowl, whisk together all ingredients.
Divide ground beef into six equal portions. Form meat into very thin patties (as thin as you can make them!). Season both sides with salt and pepper.
Heat a large skillet (preferably cast iron) over medium-high heat. Working in batches, add patties to skillet. With a spatula, press down on patties to smash them even more. Spread a tablespoon of yellow mustard onto the raw side of each patty. When bottoms are seared and crusty, flip patties, about 4 minutes.
Top each patty with a slice of cheese and cook until cheese is melty and burgers are cooked through, about 4 more minutes. Repeat with remaining burgers. Top each cooked burger patty with caramelized onions.
Spread sauce onto each bottom buns then top with lettuce, a tomato slice, two burger patties, and the top burger bun.
