Preheat oven to broil. Spread the buns with 2 tablespoons of butter. Season with salt and pepper. Place under the broiler until golden brown. Set aside.
Add 2 tablespoons of butte to a large cast iron skillet set over a medium heat. Add 6 cups of onions. Use a wooden spoon to toss the onions until they start to soften, about 2-3 minutes.
Reduce the heat to medium-low and cook onions until they are soft and caramelized, which takes anywhere from 25 to 40 minutes. Be sure to stir every 7 minutes or so. Once the onions are almost completely caramelized, add the salt, garlic and 1/4 teaspoon thyme. Cook another 2-3 minutes until the garlic is softened.
Increase the heat to medium and add white wine. Simmer until reduced and almost evaporated, about 2 minutes. Add 1/4 cup beef stock and reduce until almost evaporated, another 3-4 minutes. Season to taste with salt and pepper. Remove onions from pan and on to a plate, cover with foil to keep warm.
While the onions caramelize, make the patties. Add ground beef, grated onion, grated garlic, remaining teaspoon salt, 1/2 teaspoon thyme, dijon and worcestershire sauce. Use hands to combine, do not over mix. Form in to four patties and set aside.
If necessary, wipe the pan you cooked the onions in clean. Heat the skillet to a medium-high heat. Once the skillet is hot, add burger patties. Cook until seared on one side, about 3 minutes. Flip and sear another 2 minutes, and then evenly divide the cheese between the patties. Add the remaining beef stock to the pan and cover with a lid. Cook the burgers another 30 seconds or so until the cheese melts. (The liquid help to keep the patties moist and melt the cheese.)
Serve patties on top of crispy buns, then pile with caramelized onions.