To make the filling, combine 3 tablespoons soy sauce, 1/2 tablespoon fish sauce (if using), 1 teaspoon hoisin sauce and the Chinese five spice powder. Add 1 clove of minced garlic, the ginger, 2 green onions, and 1 tablespoon sesame oil in a Ziploc bag or bowl. Shake well to combine. Place the chicken in the bag, press out the excess air, and seal. Refrigerate and marinate for at least 30 minutes, or up to 3 hours.
Heat 2 tablespoons of sesame oil in a wok or large skillet over high heat. When the oil is hot, add the chicken + marinade (being careful of splattering) to the pan and cook until the chicken is cooked throughout and lightly browned, about 4-5 minutes. Transfer to a plate and set aside.
Add the remaining tablespoon sesame oil to the pan. Add the shredded brussels sprouts, carrots and mushrooms. Add the remaining tablespoon of soy sauce, fish sauce and remaining teaspoon of hoisin sauce. Stir to toss the vegetables. Cook over high heat for about 8-10 minutes. Toss in the remaining garlic and green onions and cook for 1 more minute. Stir the chicken into the veggies.
To wrap the egg rolls, lay an egg roll wrapper out on your work surface with one corner pointing toward you. Add a tablespoon or two of the filling to the center of the wrapper. Fold the corner closest to you up over the filling. Roll slightly away from you then fold in the two outside corners, and continue rolling the wrapper tightly away from you. Securely close the egg roll by lightly brushing the inside top corner of the wrapper with the beaten egg. Repeat with the remaining wrappers and filling.
To fry the egg rolls, heat about 3 inches of oil to 375 degrees F. Fry a few egg rolls at a time in the hot oil but without overcrowding, about 4-5 minutes or until golden brown. Transfer to a large plate lined with paper towels to help drain the excess grease. Repeat with the remaining rolls.
To bake the egg roles, preheat the oven to 425 degrees F. Spray tops of egg rolls with cooking spray. Bake for 10-15 minutes or until lightly browned.