Preheat the oven to 400°F.

Make the dressing by blending all the ingredients together in a food processor or blender.

Salt the steak on both sides. In a medium skillet over high heat, heat 1 tablespoon of olive oil. Once oil is hot, cook steak until browned and seared, about 3 1/2 minutes per side—a total of 7 minutes for medium rare (or, if preferred, about 6 minutes total for rare and 8 minutes total for medium). Set aside and allow to rest.

On a baking sheet in the oven or in the toaster oven, toast the bread until light brown, about 5 minutes. Remove from the oven and add two slices of provolone to each slice of bread and 1 tablespoon of parmesan on half of the slices of bread. Return the slices to the oven until cheese is melted and bubbling, about 3 minutes.

Make the sandwiches: Spread a generous amount of the horseradish dressing over the slices of bread. Using a sharp knife, slice the steak across the grain in ¼-inch strips and layer them over the dressing. Add the romaine lettuce,  avocado, and marinated onion plus a drizzle of the onion oil. Season with a pinch of salt, close the halves together and enjoy these delicious juicy steak sandwiches!