Place olive oil, butter, brown sugar and onions to an 8" nonstick skillet.
Stir to combine and cook over a low flame until onions are tender and caramelized, about 15 minutes.
Remove from heat and set aside.
Philly Cheesesteak
Preheat oven to 200 degrees.
Slice sub rolls lengthwise down the center and place on a baking sheet.
Toast in oven until slightly golden, about 2 minutes. Remove from oven and set aside.
Place a paper towel over a plate and set aside.
Slice steak with the grain into 1/4 inch strips, then chop into pieces.
Add vegetable oil to a 12-inch non stick pan. Preheat over a medium high flame until the oil is smoking.
Carefully place meat in pan, quickly separating evenly in the pan with a wooden spoon. Sprinkle salt and pepper over top. Do not stir.
Allow meat to brown without stirring for 5 minutes.
After 5 minutes, stir and allow to finish cooking for an additional 2 minutes.
Place cooked meat on paper towel lined plate and wipe the skillet clean of rendered fat.
Return meat to pan and sprinkle evenly with grated parmesan cheese.
Evenly lay out slices of cheese singles over meat.
Add beef broth and continue heating, folding in melted cheese into the meat to combine.
Divide meat evenly over toasted rolls. Spoon caramelized onions over top of mean. Place a slice of provolone over top of meat and onions.
Place sandwiches back in the 200 degree oven and warm until the top layer of cheese melts and bun are warm. About 2 minutes.
Serve warm.
Notes
The meat and onions can be cooked a day ahead and stored in the refrigerator. When ready to assemble, toast the buns, add the meat, onions and top layer of cheese and place in oven until cheese has melted and the meat and buns are warmed.