In a large bowl, combine all ingredients (chicken, broccoli, garlic, onion, cheese, soy sauce, salt, and black pepper). Mix together well so that all ingredients are well combined.
For the Rolling & Frying:
Using egg roll wrappers or spring roll wrappers, add about 2 Tablespoons of chicken filling to each wrapper. Tuck and roll the wrapper around the filling (watch the video in the post above to see how to roll even and tight rolls). Brush the top corner of the wrapper with water or beaten egg to help seal the wrapper end, and then finish rolling the egg roll. Repeat for all the egg rolls.
Fill a large frying pan with about 1/2-inch of vegetable and heat oil to about 350°F.
Slowly add egg rolls to the frying pan, without crowding the egg rolls in the pan because they will stick together if placed too close
Fry each side of egg roll for about 3 minutes or until crisp and golden brown. Remove and blot on a paper towel to remove excess oil.
Serve warm with hoisin-peanut dip.
You can also wrap these chicken vegetable egg rolls in lettuce and them eat them as lettuce egg roll wraps!