In a glass bowl or a plastic resealable freezer bag, whisk the Italian dressing, the Worcestershire sauce, the garlic, the salt, and the pepper together.
Add the steak to the marinade and turn the meat to coat.
Marinate the steak in the fridge for at least 30 minutes and up to overnight.
In a large skillet over medium-high heat, heat 1 tablespoon of the vegetable oil until it is smoking.
Pat the excess marinade off the steak and add it to the heated skillet and cook, undisturbed for 1 minute.
Cook the steak, while stirring, until it reaches an internal temperature of 145 degrees F, about 1-2 minutes.
Transfer the cooked meat to a fine-hole strainer to remove any excess liquid by pressing down on the meat.
In the same skillet, heat 1 tablespoon of the olive oil over medium-high heat.
Add the mushrooms to the skillet and sauté until they are golden, about 3-5 minutes. Transfer them to a paper-towel-lined plate.
To the same skillet, heat the remaining olive oil over medium-high heat.
Add the peppers, the onions, and the jalapeños to the skillet and cook until softened, about 6-8 minutes.
Add the pepper mixture and the cooked steak to a food processor and pulse until the mixture is roughly chopped.
In a small bowl, combine the provolone cheese and the cheddar cheese.
Position an egg roll wrapper on a flat surface with one point towards you.
Line the center of the egg roll wrapper with 3 tablespoons of the cheese mixture and top it with 1/4 cup of the steak mixture.
Fold the bottom corner of the wrapper over the filling then continue to roll it up, firmly folding the sides toward the center over the filling as you roll.
Seal the edges of the wrapper with the egg wash.
Place the finished egg roll on a baking sheet that is lined with parchment paper and cover with a damp towel to prevent it from drying out.
Repeat the egg roll process until all of the ingredients have been used.
In a stock pot or deep-fat fryer heated to 350 degrees F or medium-high heat, heat the remaining vegetable oil.
Fry the egg rolls in batches of 3-4 at a time, turning the rolls a few times while frying, until golden-brown, about 3-4 minutes.
Drain the fried egg rolls on paper towels.
Serve warm with the queso blanco sauce.