Place ribeye in the freezer for 25 minutes to firm it slightly, then slice very thinly across the grain; the freezer time is included in prep time. Keep sliced beef refrigerated while cooking vegetables.
Heat 1 tablespoon oil in a 12-inch skillet over medium-high heat. Cook onion, pepper and mushrooms with 1/4 teaspoon salt for 10 to 12 minutes, stirring occasionally, until their liquid evaporates and their edges brown; transfer to a bowl.
Add remaining oil to the skillet. Cook beef in 2 batches for 2 to 3 minutes per batch, spreading it thinly and turning once, until each batch reaches 145°F. Transfer cooked beef to a clean plate.
Return vegetables and all beef to the skillet with remaining salt, pepper and Worcestershire. Toss over medium heat for 1 minute, then spread evenly and scatter white American over the top.
Cover and heat on low for 1 to 2 minutes until cheese melts. Remove from the heat and rest for 3 minutes before dividing among 4 plates.