In a large sauté pan, heat olive oil over medium. Add onions, stir to coat, then cook, stirring occasionally, until starting to soften but not brown, about 10 minutes. (Depending on the size of your pan, you may need to add onions incrementally as the onions cook down and space in the pan becomes available.)

Add salt and stir to coat then cook, stirring occasionally, until onions soften, about 10 minutes. Add sugar and stir to coat, continue cooking until jammy and deeply golden brown, about 30 minutes. You may need to add water, by tablespoon measure, to keep onions from burning and sticking to pan.

Once caramelized, add bourbon and Worcestershire sauce and cook until liquid has evaporated, about 2 minutes. Remove from heat and season with salt and pepper to taste.

In a medium bowl, whisk together mustards, mayonnaise and vinegar; season with salt and set aside.

Divide meat into quarters (about 4 oz. each) and roll into balls.

Working with 1 ball at a time, place ball on a piece of parchment paper, top with a second piece of parchment paper and press into a ¼-inch thick patty using a burger press or iron. Repeat with remaining ground beef balls; layer pressed patties between parchment to avoid sticking together.

Heat a large cast-iron, stainless steel, or carbon steel skillet (or flat top) over medium-high heat until ripping hot and smoking, about 5 minutes.

Toast buns, cut side down, in skillet until lightly golden, about 1 minute.

Increase heat to high. Season one side of patties with salt, place patties salted side down, in skillet; season generously with salt and cook, without moving but smashing lightly with burger iron to keep flat, until bottom sides are crisp and browned, about 1½ minutes.

Flip patties then top with caramelized onions and a slice of cheese. Cover and cook until cheese is melted, about 1 minute.

Wipe out skillet with paper towels before repeating cooking process with remaining beef patties, onions and cheese.

To assemble, spread 1 tablespoon burger sauce on both cut sides of buns (tops and bottoms). Place patty on bottom bun then top with top bun and serve.