Trim any excess fat from the skirt steak. Slice the steak crosswise into 3-inch-wide strips, place on a large plate, and freeze until very firm, about 1 hour.
Using a sharp knife, shave the frozen steak as thinly as possible against the grain. Pile the sliced meat on a cutting board and chop coarsely 10 to 20 times to break it up.
Preheat the oven to 400°F and position the rack in the middle. Split the rolls and place them cut-side up on a baking sheet; toast in the oven until lightly browned, about 5 to 10 minutes.
Heat 1 tablespoon oil in a 12-inch nonstick skillet over high heat until just smoking. Add half the chopped meat in an even layer and cook without stirring until well browned, about 4 to 5 minutes. Stir and cook until no longer pink, 1 to 2 minutes, then transfer to a colander to drain.
Wipe out the skillet with paper towels. Add the remaining 1 tablespoon oil and cook the second half of the meat the same way, then drain and return all the meat to the skillet.
Return the skillet to medium heat, drain any excess moisture, and add 1/2 teaspoon salt and 1/8 teaspoon pepper. Cook, stirring constantly, until the meat is heated through, 1 to 2 minutes.
Reduce heat to low, sprinkle the meat with the grated Parmesan, and layer the American cheese slices over the top. Let the cheeses melt, about 2 minutes, then fold the melted cheese into the meat until evenly combined.
Divide the cheesy meat mixture evenly among the toasted rolls and serve immediately.