Heat oil in a large nonstick skillet over medium-high heat. Add onion and cook, stirring often, until soft and beginning to brown, 2 to 3 minutes. Add mushrooms, bell pepper, oregano and pepper and cook, stirring often, until the vegetables are wilted and soft, about 7 minutes. Add steak and salt and cook, stirring, until the meat is just cooked through, about 4 minutes.

Reduce heat to low. Sprinkle the vegetables and meat with flour; stir to coat. Stir in banana peppers (if using) and broth; bring to a simmer. Remove from the heat; lay cheese slices on top of the vegetables and meat, cover and let stand until melted, 1 to 2 minutes.

Divide into 4 portions with a spatula, leaving the melted cheese layer on top. Scoop a portion onto each bun and serve immediately.

Kitchen tip: It is easiest to cut meat into thin slices if it is partially frozen. If you have time, freeze it for 20 minutes before slicing.