In a bowl, add tuna, mayonnaise, gherkins, celery, onion, salt, black pepper, red pepper flakes, dill, and parsley.
Mix everything together, using a fork to break up the tuna into smaller pieces. It should be fairly crumbly and dry.
To assemble, place an egg roll wrapper on a dry board, angled so one of the corners is pointing down towards you. Take the bottom corner and fold it ⅓ of the way to the top of the egg roll wrapper.
Place the tuna mixture and the cheddar cheese stick on the folded corner.
Fold the right corner of the egg roll wrapper towards the center, so it is on top of the tuna and cheese mixture. Do the same for the left side.
Pinch the bottom of the egg roll wrapper while you begin tightly rolling the egg roll wrapper up until there is an inch of egg roll wrapper remaining.
Mix together the egg and water to make an egg wash and brush the egg wash on the top corner of the egg roll wrapper. Continue rolling to seal shut.
Generously spray or brush oil onto all of the egg rolls and air fry at 400F for 5-6 minutes until golden and crispy, flipping halfway.
Make the sauce by mixing together the mayonnaise, pickle or lemon juice, ketchup and diced gherkins. Top the egg rolls with Parmesan cheese, as desired and enjoy immediately with sauce!