Slice the onions in half and thinly slice. Place in a bowl and mix in 1 teaspoon of salt. Let sit for 15-20 minutes. The onions will begin to release liquid. Place on a tea towel and squeeze out as much liquid as possible. Set Aside.
While the onions are doing their thing, let's make the mushrooms! Finely chop the mushrooms and place in a medium-low preheated (cast iron skillet, if available). Melt the butter and add the mushrooms. Cook for about 5 minutes until the mushrooms begin to brown. Season with a pinch of salt and pepper and remove from the pan and set aside to cool.
In the same pan, melt another 2 tablespoons of butter over medium-low heat. Add the onions and cook until they become translucent and begin to brown. Remove from pan and set aside.
When the mushrooms have cooled and the onions are cooking, let's make the hamburgers. In a large bowl combine the hamburger, mushrooms, garlic, Worcestershire, 1/4 teaspoon salt and 1/4 teaspoon pepper. Combine well. Form into 4 equal size balls and set aside.
In the same pan that the onions cooked in melt another 2 tablespoons of butter over medium heat. Depending on the size of your skillet(I had to do 2 batches) place the hamburger ball. Smash down with a heavy spatula (I used the one from my grill). So it's about 1/2 inch thick. Place about 1/2 cup of onions on top of each burger and cook for 3 minutes. Flip the burger over so that the onions are on the bottom. Place a piece of cheese or two on each burger. When the cheese begins to melt, place the buns on top of the cheese like this. Cover with a lid and steam the buns for one minute or so.
Carefully remove the bottom bun, and slide one burger with onions on top of the burger bun. Top with the other bun and serve!