Heat 2 tablespoons oil in a large fry pan over medium heat. Add onions and peppers and saute until peppers are soft and onions are translucent, about 5 minutes. Sprinkle shaved beef with salt and add to pan. Add Worcestershire sauce and cook until meat is no longer pink, about 5 minutes. Turn off heat and adjust seasonings. Transfer mixture to a fine mesh sieve and strain to remove any excess liquid and let cool slightly.
Lay out one eggroll wrapper with the corner pointed to you. Using a pastry brush or your finger, lightly wet the edges of the entire wrapper with water. Spread 1 tablespoon cheese sauce (or place 1 piece provolone) on the lower third of the wrapper and top with 1 to 2 heaping tablespoons of the filling. Roll the point facing you over the filling, keeping the eggroll tight. When you reach the middle of the wrapper, fold in the right and left sides, then continue to roll until you have a tight eggroll. Press to seal the wrapper to itself, adding more water to seal the final edge if needed. Repeat with remaining wrappers and filling.
Heat about 3 inches oil in a heavy bottomed pot to 375 degrees F. Add rolls in batches and cook, flipping occasionally, until golden brown, 3 to 4 minutes. Drain on paper towels and serve with Pepperoncini Mayo Dipping Sauce.
Pepperoncini Mayo Dipping Sauce:
Add mayonnaise, pepperoncini and pepperoncini liquid, mustard, vinegar and sugar to a food processor and process until well combined. Transfer mixture to a serving bowl.