Toss the onions, peppers and mushrooms in a bowl with 1 tablespoon of the oil and some salt and pepper.
Heat a large cast-iron griddle over medium heat. Spread the butter onto the griddle and place the rolls cut side down. Lightly toast the buns on the griddle, about 2 minutes. Put the buns on a baking sheet until ready to fill.
Add 1 to 2 tablespoons more oil to the griddle. Using long metal tongs, spread the onions, peppers and mushrooms across the hot griddle. Cook the vegetables until they start to blister and soften, about 4 minutes. Push the vegetables to one side of the griddle and put the steak down in a single layer on the other side. Sprinkle the steak with a large pinch of salt and pepper. Add a little more oil if needed. Cook the steak until it is no longer pink, about 4 minutes. Then put the pastrami on top. Cook all together until the steak edges are crispy and caramelized and the pastrami is warmed through.
Turn off the heat and pile the cooked vegetables onto the bottom of the rolls. Separate the meat into two piles and place on top of the vegetables. Top each sandwich with 3 slices of cheese per bun. Put the baking sheet in the oven until the cheese is melted and bubbling, about 2 minutes. Serve immediately.