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Preheat the oven to 425°F with a rack in the middle position.
In a large bowl, mix the olive oil, mustard, Worcestershire, salt, pepper, garlic powder, onion powder, and brown sugar until combined. Add in the steak, onions, bell peppers, and mushrooms (if using) and turn to coat.
Spread out the steak and veggie mixture on a rimmed sheet pan and roast until the veggies are softened and the steak is cooked through, about 15 minutes.
Meanwhile, make the cheese sauce. In a small saucepan, melt the butter over medium heat. Sprinkle the flour over the butter and stir continuously until a thick paste forms, about 1 minute.
Gradually pour in the milk, stirring, until incorporated. Add the mustard, provolone, garlic powder, onion powder, Worcestershire, and salt, whisking until the cheese has melted and the sauce has thickened, about 5 minutes more.
Remove the sheet pan from the oven and turn the oven to broil. Open the hoagie roll and toast them directly on the middle rack until lightly browned, 1-2 minutes.
Divide the steak and veggies between the hoagie rolls, then drizzle the cheese sauce and serve.