Preheat a large cast-iron skillet to medium hot. Take a few paper towels and fold them several times to make a thick square. Blot a small amount of oil on the paper towel and carefully and quickly wipe the hot skillet to make a nonstick surface.
Season the ground meat in a bowl with salt and pepper. Give it about 3 turns in the bowl with your hands or a big spoon and it's done. Shape into 4 patties. When the skillet is hot, put the burgers on the skillet and cook 4 minutes on the first side and 5 for the second side for rare; 7 minutes per side for medium, adding the cheese during the final minute or 2 to melt. Remove the burgers to a plate. Rub the skillet with the folded paper towel again to clean it. Then toast the buns cut side down for about 1 minute, just to mark them. Serve the hamburgers in the buns with the accompaniments below.
Saute 1 pound thickly sliced mushrooms in 1/4 cup olive oil with 2 sliced garlic cloves and 1 tablespoon fresh thyme leaves over super-high heat until nicely browned, 8 to 10 minutes. Stir in a couple of drops fresh lemon juice to brighten the flavor.
Heat 2 tablespoons unsalted butter with 2 tablespoons olive oil in a large pan over medium heat. Add 2 onions, cut into 1/2-inch-thick slices, sprinkle with salt and pepper and cook slowly until well caramelized, 15 to 20 minutes.
Bacon with Rosemary:
Lay out 1/2 pound sliced bacon on a baking sheet, strip leaves from 1 sprig rosemary and throw them on top with lots of cracked black pepper and roast at 400 degrees F until the bacon is crisp, about 10 minutes.
Herbed Horseradish Mayonnaise:
Stir together 1 cup mayonnaise, 1 tablespoon prepared horseradish, 1/4 cup freshly minced herbs such as parsley, basil, chives or scallion, 2 tablespoons olive oil, a little squeeze of lemon juice, and salt and pepper.
Thickly slice 2 pounds red and yellow beefsteak tomatoes, drizzle with olive oil, and sprinkle with sea salt and 1/2 bunch chopped fresh chives.