Heat a large, fluted grill pan over medium-high heat until very high.
Cut the onion into 6 slices, without cutting off the tail-base, so that the onion does not crumble. Place the onion in a bowl, add oil, sprinkle with 1/2 teaspoon of salt and 1/4 teaspoon of pepper, and stir.
Put the onion wedges in a preheated pan, fry for about 5 minutes on one side and 4-5 minutes on the other.
Sprinkle the steak on all sides with 1 teaspoon of salt and 1/2 teaspoon of pepper. Put the meat in a pan and fry for about 5-6 minutes on each side (the temperature on the thermo for meat should reach 70 degrees). Transfer the steak to a board, cover with foil. Transfer the bow to the board as well.
Cut the buns in half and put in slices in a pan, fry until lightly charred.
In a small bowl, combine the cheese, mayonnaise, horseradish and cayenne pepper. Grease the slices of buns with this sauce.
Cut the base tails from the onion slices. Cut the steak into strips. Place the meat, onion and lettuce between the buns. Serve the sandwiches immediately.