1 ½ pounds top round steak, patted dry

8 (3/4-ounce) American cheese slices, torn

6 (2/3-ounce) provolone cheese slices, torn

1 medium-size (10 ounces) sweet onion, chopped (about 1 3/4 cups)

1 medium-size (8 ounces) red bell pepper, sliced (about 1 1/2 cups)

1 medium-size (8 ounces) green bell pepper, sliced (about 1 1/2 cups)

4 crusty Italian hoagie rolls, split horizontally, keeping 1 long edge intact

Place steak on a small baking sheet lined with parchment paper. Freeze, uncovered, until partially frozen, about 45 minutes.

Meanwhile, whisk together cornstarch and 1/4 cup of the milk in a small saucepan until smooth. Whisk in remaining 3/4 cup milk. Bring to a simmer over medium, whisking occasionally. Simmer, stirring occasionally, 1 minute. Reduce heat to low. Gradually add American and provolone cheeses, whisking constantly, until melted, about 2 minutes. Remove from heat. Cover cheese sauce to keep warm.

Halve steak crosswise, then cut against the grain into very thin (about 1/6-inch-thick) slices. Place in a medium bowl; set aside.

Heat a griddle pan over medium-high or an electric griddle to 400°F. Add onion, 1 tablespoon of the oil, and 1/4 teaspoon of the salt. Cook, stirring often, until onion is browned, 6 to 8 minutes. Transfer to a small bowl. Do not wipe griddle clean. Add 1 tablespoon of the oil to griddle. Add bell peppers, 1/2 teaspoon of the salt, and 1/4 teaspoon of the black pepper to griddle. Cook, stirring occasionally, until bell peppers are very tender, 15 to 20 minutes, adding 2 tablespoons water at a time as needed to prevent excess browning. Stir mixture into onion in bowl. Wipe griddle clean.

Pull apart steak in bowl using tongs; add remaining 1 1/2 teaspoons salt and 1/2 teaspoon black pepper, and toss to coat. Add remaining 1 1/2 tablespoons oil to griddle. Add steak in an even layer; cook, undisturbed, until bottom side is browned but top is still pink, 2 to 3 minutes. Stir steak; cook, undisturbed, until no more pink remains in steak, about 1 minute. Turn off griddle, and push steak to 1 side of griddle.

Reheat cheese sauce over medium, stirring occasionally, until hot. Spread 2 tablespoons cheese sauce onto cut sides of each hoagie roll. Divide steak and bell pepper mixture evenly among rolls; top each with 1/4 cup cheese sauce.