Place the ribeye in the freezer for 20 minutes to firm up, then slice as thinly as possible against the grain. Slice the pepper and onion into thin strips. Divide the ground beef into 4 equal portions and gently form into patties slightly wider than your buns; season both sides with salt and pepper.

Heat the oil in a large skillet over medium-high heat until shimmering. Add the peppers and onions with a pinch of salt and cook, stirring occasionally, until softened and browned at the edges, about 8 minutes. Push to one side of the pan.

Add the sliced ribeye to the empty side of the pan, breaking it up with a spatula. Cook for 2 to 3 minutes until browned but still juicy. Mix with the vegetables, add the Worcestershire sauce, and toss to combine. Reduce heat to low and top with 4 slices of provolone. Cover the pan briefly to melt the cheese, about 1 minute. Keep warm.

Preheat your grill or grill pan to medium-high heat. Cook the patties for 4 minutes per side for medium, or until an internal thermometer reads 160F. In the last minute of cooking, top each patty with a slice of provolone and close the lid or tent with foil to melt.

While the burgers rest, spread a thin layer of mayonnaise on the cut sides of the buns and toast cut-side down on the grill or in a dry skillet until golden, about 1 minute.

Place a cheese-topped patty on each bottom bun. Pile a generous portion of the cheesesteak mixture on top, letting any melted cheese drip down. Cap with the top bun and serve immediately while hot.