Preheat oven to 375 degrees F and add vegetable oil to a large baking sheet, enough to cover the whole pan in a thin layer.
Next, bread the chicken. In a shallow dish, whisk together egg and water and set aside. Fill a separate shallow dish with the breadcrumbs and set aside. Then, working one piece at a time, coat the chicken fully in the egg wash, then the breadcrumbs, and transfer to the prepared baking sheet.
Bake chicken until crispy and cooked through, about 20 to 25 minutes, flipping halfway to crisp both sides.
Remove chicken from the oven and lower heat to 350 degrees F. Top each piece of crispy chicken with a generous amount of tomato basil sauce, then a layer of shredded mozzarella cheese. Bake for 10 minutes to warm the sauce through and melt the cheese.
Slice sub rolls and place them open-faced on a baking sheet for toasting. Once the chicken parm is done baking, switch oven to broil on HIGH. Toast sub rolls on the bottom rack of the oven while the chicken parm browns under the broiler, about 3 minutes.
Finally, assemble the sandwiches. Transfer 1 to 2 pieces of chicken parm to each sub roll, then sprinkle with grated parmesan cheese before closing. Serve immediately.