Step 1

Place the meat in the freezer for 35 to 45 minutes (having the meat partially frozen makes it easier to slice). Using a sharp knife, slice the steak as thinly as possible against the grain. (If necessary, cut the steak in half horizontally first to create manageable pieces. This helps ensure the slices aren't too long.) Mound the meat on the cutting board and chop it coarsely with a knife 15 to 20 times. Season the meat with ¾ teaspoon salt and ½ teaspoon pepper.

Step 2

Preheat the oven to 400°F and set an oven rack in the middle position. Open the rolls and arrange on a baking sheet, face-up. Toast until lightly browned, 5 to 8 minutes. Cover to keep warm.

Step 3

Meanwhile, heat 1 tablespoon of oil in a large cast iron or nonstick skillet over medium-high heat until it shimmers. Add the onions, peppers, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, stirring frequently, until softened and golden brown, 10 to 12 minutes. Stir in the vinegar, then transfer the mixture to a large plate.

Step 4

Wipe the skillet clean with a paper towel. Heat another tablespoon of oil over high heat until almost smoking. Add half of the meat in an even layer and cook without stirring until well browned on one side, about 2 minutes. Stir and continue to cook until the meat is no longer pink, about 1 minute more. Using a slotted spoon, transfer the meat to the plate with the vegetables.

Step 5

Add the remaining meat to the skillet in an even layer, adding more oil only if necessary. Cook without stirring until well browned on one side, about 2 minutes. Stir and continue to cook until the meat is no longer pink, about 1 minute more.

Step 6

Return the first batch of cooked meat and vegetables to the skillet, and add the American cheese. Heat, stirring constantly, until the meat is warmed through and the cheese is melted and evenly distributed, about 1 minute. Reduce the heat to low and shingle the provolone cheese over the meat. Allow the cheese to melt, about 2 minutes. Use a spatula to divide the mixture evenly among the toasted rolls. Serve immediately.