In a large skillet over medium heat, heat 1 tablespoon of oil for about 30 seconds, then add onion and garlic. Cook for 2 minutes.
Add ground beef, Worcestershire sauce, salt, and pepper. Cook until beef is no longer pink, for about 5 minutes.
Remove pan from heat and drain.
Mix mustard and ketchup into the beef mixture and allow it to cool slightly for about 10 minutes. Set aside.
In a large, heavy-bottomed pot over medium-high heat, add the remaining 2 cups of vegetable oil. It should measure to a depth of about 1". Heat until the oil reaches 350°F.
Line a large plate with paper towels and set it aside.
Lay 1 egg roll wrapper with a point facing toward you, like a diamond shape.
Spoon 2 tablespoons of the beef mixture onto the middle of the wrapper.
Sprinkle 1 tablespoon of cheese onto the meat.
Brush the egg onto the opposite two sides of the wrapper.
Roll up like a burrito, making sure all filling is sealed inside.
Repeat with remaining wraps and fillings.
Place 4 rolls in hot oil and cook 2 minutes per side, until golden brown.
Remove to the lined plate. Repeat with the remaining rolls.
Serve and enjoy!