Pour the oil into a 10 inch straight sided skillet over medium heat. Warm until it reaches 350 degrees.
Preheat the oven to 400 degrees.
Place the chicken cutlets on a board and cover with plastic wrap. Using a meat mallet, lightly pound the cutlets until 1/4 inch thick. Remove the plastic wrap and season the cutlets evenly using 1/2 teaspoon salt and 1/2 teaspoon pepper. Sprinkle each cutlet with 2 tablespoons mozzarella and 2 teaspoons parmesan. Place 3 basil leaves over the top of each. With the chicken cutlet laying lengthwise, carefully roll each cutlet and secure with a toothpick.
Place the flour, egg and breadcrumbs in 3 separate bowls. Season the flour with 1/2 teaspoon salt and a 1/4 teaspoon pepper. Season the breadcrumbs the in the same manner. Working with one rolled cutlet at a time, dip the roll into the flour, the egg, then the bread crumbs, back to the egg and into the breadcrumbs to finish. Repeat until all of the rolls are completely breaded. In two batches, gently lower 4 rolls into the preheated oil and fry for 3 minutes per side or until golden brown and crispy all around. Using a slotted spoon, remove from the oil and drain on a paper towel lined plate. Place the fried rolls on a rimmed baking sheet and finish in the oven for approximately 5 minutes or until and internal thermometer reads 160 degrees. Allow to the rolls to rest for 5 minutes.
To serve, place a 1/2 cup of warmed tomato sauce on each plate. Remove the toothpicks from the rolls and place 2 rolls on each pool of marinara. Top with more grated parmesan and a sprinkling of chopped parsley if using. Serve hot.