Argentina knows beef. And Argentines know how to make it more delicious than it has a right to be. This ravishing beast of a sandwich is their go-to street snack (yes, it’s a snack), but feel free to serve it as a proper meal. Then plan on a siesta.
Place the steaks between layers of plastic wrap. Using the flat side of a meat mallet or a rolling pin, evenly pound the steaks until they are very thin. They may tear a little, and that’s ok, but try to get them an even thinness all around without falling apart.
Heat a 12-inch cast-iron pan or the flat side of a grill pan over medium-low heat for 3 minutes. Then raise the heat to medium-high for 2 more minutes. (This gets the entire surface more evenly hot without scorching the food.)
Meanwhile, grate the flesh of the tomato halves on the large holes of a box grater onto the top half of the rolls. Pound the steaks between plastic wrap until very thin.
Pour 1 teaspoon of the oil into the pan and then add the onion. Sprinkle with a little salt and pepper and stir. Cook until the onions char and soften for about 5 minutes. Remove to a plate. Add 1 more teaspoon of oil. Sprinkle the steaks liberally on both sides with salt and pepper and place in the pan. Cook, flipping, until charred and just cooked through, about 2 minutes. Turn off the heat and break the cheese into chunks and put it on top of each steak. Cover with a lid or overturned metal bowl to melt.
Meanwhile, heat a medium nonstick skillet over medium heat. Drizzle in the remaining teaspoon of oil. Crack the eggs, sprinkle with a little salt and pepper, and cook until the whites set about 3 minutes. Break the yolk if desired and flip the eggs, cooking until the yolks are not dripping, but not hard-cooked either, about 2 minutes.
Spread 2 tablespoons of mayonnaise on the bottom roll. Grind a little black pepper on the mayo. Place a steak with cheese on the mayo. Then half the onions, 4 slices of the Coppa, an egg, and a piece of lettuce, if using.