Step 1

Place steak pieces on large plate or baking sheet and freeze until very firm, about 1 hour.

Step 2

Meanwhile, adjust oven rack to middle position and heat oven to 400 degrees. Spread split rolls on baking sheet and toast until lightly browned, 5 to 10 minutes.

Step 3

Using sharp knife, shave steak pieces as thinly as possible against grain. Mound meat on cutting board and chop coarsely with knife 10 to 20 times.

Step 4

Heat 1 tablespoon oil in 12-inch nonstick skillet over high heat until smoking. Add half of meat in even layer and cook without stirring until well browned on 1 side, 4 to 5 minutes. Stir and continue to cook until meat is no longer pink, 1 to 2 minutes. Transfer meat to colander set in large bowl. Wipe out skillet with paper towel. Repeat with remaining 1 tablespoon oil and sliced meat.

Step 5

Return now-empty skillet to medium heat. Drain excess moisture from meat. Return meat to skillet (discard any liquid in bowl) and add salt and pepper. Heat, stirring constantly, until meat is warmed through, 1 to 2 minutes. Reduce heat to low, sprinkle with Parmesan, and shingle slices of American cheese over meat. Allow cheeses to melt, about 2 minutes. Using heatproof spatula or wooden spoon, fold melted cheese into meat thoroughly. Divide mixture evenly among toasted rolls. Serve immediately.