Step 1Cut the onions into the thinnest possible slices, using a mandoline or a very sharp knife. (You should have about 5 1/2 cups sliced onions.)
Step 2Put 2 tablespoons oil in a heavy skillet and heat over medium-high heat for 5 minutes. Loosely gather a handful of meat (amounting to one-sixth of the total) and slip it into the pan. Repeat with the rest of the beef to make 6 burgers, frying in batches if necessary. Burgers should be free-form lumps, not neat patties; using the side of the spatula, shape them roughly round as they cook.
Step 3After 1 minute of frying, pile one-sixth of the onions onto each burger and sprinkle with half of the salt. With as much force as you can manage, smash the onions into the meat with the spatula. Again work the burgers into a roughly round shape with the spatula and fry 1 minute.
Step 4Flip the burgers and mash them hard with the spatula. Drain some of the grease from the pan. Sprinkle the remaining salt on the burgers. Put a cheese slice on each burger and cook until the cheese melts and you can smell the onions starting to burn, about 1 1/2 to 2 minutes. (The burgers may be rather greasy; if necessary, hold with the spatula against the side of the pan for 20 seconds to let them drain.) Serve on buns brushed with mustard.