Heat the oil in a large skillet over medium-high heat.
Add the sliced onion and green bell pepper and cook, stirring occasionally, until the onion is soft and translucent, about 5–7 minutes.
Season the vegetables with the kosher salt and black pepper and stir to combine.
Push the vegetables to one side of the pan and add the thinly sliced rib eye to the empty side.
Cook the steak slices until browned, stirring as needed to separate the pieces, about 3–5 minutes.
Mix the cooked steak with the onions and peppers in the pan and divide the mixture in half.
On one half of the mixture in the pan, place three slices of provolone, then layer the second half of the meat mixture on top and add the remaining three slices so the cheese is on top.
Cover the pan with a bowl for 1–3 minutes until the provolone melts.
Spread one quarter of the melted meat and vegetable mixture onto a flatbread, leaving a 1/2-inch border around the edges.
Starting at one end, roll the flatbread tightly to enclose the filling, then slice the wrap in half and serve immediately.