Place the chicken on a large plate and freeze until slightly firm, about 1 hour. Using a sharp knife, shave the frozen chicken as thinly as possible against the grain. Mound the meat on a cutting board and chop coarsely with your knife, 10 to 20 times.
Adjust the oven rack to the middle position and heat oven to 400℉. Split rolls open, spread on baking sheet, and toast until lightly browned, 5 to 10 minutes.
Heat 1 tablespoon oil in 12-inch nonstick skillet over high heat until just smoking. Add half of the chicken in an even layer and cook, without stirring, until well-browned, 4 to 5 minutes. Stir and continue to cook until the chicken is no longer pink, 1 to 2 minutes; transfer the meat to a colander to drain. Wipe out the skillet with paper towels and repeat with the remaining 1 tablespoon oil and chopped chicken.
Return the now-empty skillet to medium heat. Drain the excess moisture from the chicken, return the chicken to the skillet, and add ½ teaspoon salt and ⅛ teaspoon pepper. Cook, stirring constantly, until chicken is warmed through, 1 to 2 minutes.
Reduce heat to low, sprinkle with Parmesan, and lay American cheese over the top. Let cheeses melt, about 2 minutes. Fold the melted cheese into the chicken thoroughly. Divide the mixture evenly among the toasted rolls and serve.