In a medium skillet over medium heat, melt one tablespoon of butter with one tablespoon of olive oil. Place the onion and pepper slices into the pan and slowly sauté for 15-20 minutes until browned and caramelized, stirring and flipping occasionally. Add a pinch of salt, a few grinds of pepper and a large pinch of sugar and stir to combine. Cook on low for another 5 minutes. Remove from heat.
While onions are cooking, heat a large skillet over high heat with one tablespoon of butter and one tablespoon of oil. (Mushrooms will be cooked in two batches). Once butter mixture is hot, place half the mushrooms in and stir. Cook for 2-3 minutes and flip each slice and cook for another 2 minutes then remove to a bowl.
Repeat with one tablespoon of oil and butter and second half of mushrooms.
When the second batch of mushrooms are cooked, add first batch back in with second batch and pull pan from burner. Add in bourbon and slowly place back on flame. The mixture should flame up and then burn off the alcohol. Season with salt and pepper and remove from heat.
Once the peppers and onions are cooked, add cooked mushrooms.
Heat large skillet or flat griddle over medium high heat and add one tablespoon of olive oil and one tablespoon of butter. Add in half the steak and with two wooden spoons or tongs, pull steak apart and turn and cook for 2-3 minutes until cooked through. Season with a little more salt and pepper. Remove to the pan with the peppers and onions.
Add remaining oil and butter and cook the rest of the beef then add to the other pan.
Stir the mixture around and heat to hot. Make four large piles and place provolone slices over each pile and cover pan to melt the cheese, about one minute.
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I have found vast differences in what markets call shaved steak. Typically what you find in the market is steak shaved off a frozen block of beef scrap pieces. They put the scraps together into a square shape, freeze it and slice it paper thin. What you get after the meat thaws out is a lot of weepy tough pieces. In my opinion, the best option for shaved steak is a rib eye that you have a butcher shave for you. It cooks up quickly and is super tender and flavorful.