Pour 1 to 2 inches of oil into a pan and heat it to 350°F. You can also use a deep fryer if you have one. Let the oil come up to temperature while you prep, so it's ready when the rolls are.
Lay out the egg roll wrappers and slice each one into four equal squares. You'll end up with 20 small squares. Keep them under a damp towel so they don't dry out and crack.
Chop the shredded mozzarella into smaller pieces. Tiny pieces fit neatly inside the little wrappers. They also melt faster, which is perfect for a 2-minute fry.
Add the chopped pepperoni, pizza sauce, and mozzarella to a bowl. Stir until everything is evenly coated. The mix should hold together without dripping.
Spoon a heaping teaspoon of filling onto the left side or center of a square. Leave about half an inch of empty wrapper above and below. Don't overfill, because extra filling is what makes rolls burst.
Dip a finger in water and wet all the edges of the wrapper. Roll it up and press the open top and bottom edges together. Wet those ends again and fold each one up to make a tight seal.
Slide about four rolls into the hot oil at a time. Fry for 1 to 2 minutes until they look puffy and golden. Don't crowd the pan, or the oil temperature will drop.
Lift the rolls out with a slotted spoon and set them on a paper towel-lined plate. Let them rest for a minute, since the cheese inside is very hot. Serve with extra pizza sauce for dipping.