In a large bowl stir together flour, onion powder, and garlic powder. Add meat pieces, stirring to coat.
In a 6- to 8-qt. Dutch oven heat oil over medium-high. Add chopped onions; cook 6 to 8 minutes or until tender, stirring occasionally. Add chopped sweet peppers; cook and stir 3 to 4 minutes or until tender. Add garlic; cook and stir 1 minute. Working in batches, add meat and any remaining flour to vegetable mixture. Cook and stir over medium heat 8 to 10 minutes or until meat is lightly browned.
Increase heat to high; add broth and Worcestershire sauce. Cook and stir until mixture begins to thicken; reduce heat slightly. Cook and stir until boiling. Remove from heat. Top with cheese; cover and let stand 1 to 2 minutes or until melted. Stir to combine.
If desired, top servings with cooked pepper strips, onion slices, mushrooms, and/or additional American cheese. Serve with toasted bun strips.
To get the most even, thin slices of steak, place steak in the freezer for about 30 minutes before slicing. The quick chill will firm the meat and make it easier to cut.