Cook onions in half the oil 6 to 8 minutes. Add peppers and mushrooms, cook 4 to 5 minutes more. Add garlic 30 seconds. Transfer to a plate.
Raise heat to high. Add remaining oil and heat until shimmering. Pat beef dry and sear in batches 90 seconds undisturbed, flip 45 to 60 seconds. Season with salt, pepper, paprika, and Worcestershire.
Return vegetables to skillet and toss with beef over medium heat 1 minute.
Lay cheese over filling. Cover and melt 60 to 90 seconds. Serve in bowls over chosen base.