1 pound skirt steak, trimmed and sliced with grain into 3-inch wide pieces
1 tablespoon olive oil, plus more for brushing
1 small onion, finely diced
1/2 large green pepper, finely diced
1 tablespoon vegetable oil
2 tablespoons grated Parmesan
4 slices Monterrey Jack cheese
Salt and pepper
30 wonton wrappers
Place steak pieces on plate in single layer and freeze for about 1 hour.
In a large skillet, heat 1 tablespoon olive oil over medium-high heat. Add onion and pepper, and cook, stirring often, until softened, about 5 minutes. Transfer to medium bowl. Wipe out skillet and set aside.
Preheat oven to 400º. Remove meat from freezer and thinly shave steak pieces against the grain.
Place skillet over high heat, and heat vegetable oil until smoking. Add meat in a single layer, and cook without stirring for 4 minutes.
Stir meat together and then drain in colander.
Reduce heat to low, return meat to skillet, and stir in Parmesan and onion and pepper mixture.
Lay cheese slices over meat, and let melt for 2 minutes.
Stir cheese into meat until completely melted and thoroughly combined. Season with salt and pepper to taste.
Transfer mixture to bowl.
Spray a large baking sheet with nonstick spray.
Set out a small bowl of water and a small bowl of olive oil. Place about 2 teaspoons of cheesesteak mixture in each wonton wrapper (I set out 6 at a time). Taking one wrapper at a time, fold sides up, dip finger in water and gently wet sides, pull top of wrapper over mixture toward you, wet edge of top of wrapper, pull bottom of wrapper up and press to seal.
Place egg rolls on prepared baking sheet and brush each one sparingly with olive oil.
Bake for 10 minutes, then flip egg rolls over using tongs.
Bake for an additional 5 to 6 minutes, until egg rolls are nicely browned on both sides. Transfer egg rolls to paper towel-lined plate to absorb any oil, and then arrange on serving platter.