In a medium saucepan, heat the oil to 325 degrees F. Line a plate with paper towels.
In a shallow bowl, whisk the eggs and milk thoroughly. In another shallow bowl, combine the panko and Italian breadcrumbs.
Dredge 1 side of an egg roll wrapper in the egg wash. Place the wrapper egg wash-side up on a work surface and put 2 salami slices in the center of the wrapper, toward the edge closest to you. Place 1 cheese stick on top of the salami. Roll the wrapper edge closest to you over the filling. Tuck in the ends and continue to roll until you have an egg roll shape.
Dredge the rolled wrapper in the egg wash again and thoroughly coat in the breadcrumb mixture. Repeat with the remaining ingredients. (The rolls can be frozen for future use at this point.)
Add 1 or 2 rolls to the oil at a time and cook 2 to 3 minutes, until golden brown. Be careful not to overcook them or cook them on too high a temperature, as the wraps may explode. Transfer the fried cheese rolls to the paper towel-lined plate and serve immediately with marinara sauce. Garnish with parmesan and parsley, if desired.