In a large bowl, stir together the Parmesan, parsley, milk, salt, oregano, pepper flakes, 1/4 cup breadcrumbs and 1 egg. Add the chicken and gently stir to combine, being careful not to overwork the meat. Shape the meat mixture into bite-size balls and press a cube of mozzarella in the middle of each, sealing it with the meat around it.
Set up a standard breading station: put the flour in one shallow bowl, the remaining 1 cup of breadcrumbs in a separate bowl and the remaining 2 eggs in another separate bowl. Roll each meatball in the flour, then the egg wash, then the breadcrumbs.
Fill a medium Dutch oven a third of the way with vegetable oil and place over medium heat. Bring the oil to 350 degrees F using a deep-fry thermometer,
Drop in the meatballs a few at a time so as to not lower the temperature, and fry until deep golden brown and cooked through, about 5 minutes. Remove to a paper towel-lined plate to drain and season with a pinch of salt. Continue with the remaining meatballs. Serve warm with marinara for dipping if desired.