Freeze ribeye 25 minutes to firm slightly, included in prep time. Thinly slice across grain; refrigerate while cooking vegetables.
Heat 1 tablespoon oil in 12-inch skillet over medium-high. Cook onion, pepper, and mushrooms with 1/4 teaspoon salt 10 to 12 minutes, stirring occasionally, until liquid evaporates and edges brown. Transfer to bowl.
Add remaining oil. Cook beef in 2 batches, thinly spread, 2 to 3 minutes per batch, turning once, to 145°F each. Transfer to clean plate.
Return vegetables and beef with remaining salt, pepper, and Worcestershire. Toss over medium 1 minute; spread evenly and scatter American cheese over.
Cover over low 1 to 2 minutes until melted. Remove from heat; rest 3 minutes and divide among 4 plates.