In a small bowl, combine the breadcrumbs and milk; set aside to let the breadcrumbs soften.
Heat the butter and olive oil in a skillet over low heat; add the chopped onion and sauté for 10–15 minutes until soft and beginning to brown, then remove from heat and let cool slightly.
Place the ground chicken in a food processor; add the soaked breadcrumbs (including any remaining milk), sautéed onion, egg, chopped parsley, minced garlic, salt, and pepper. Pulse until ingredients are evenly combined but not pureed.
Pour vegetable oil into a large skillet to a depth sufficient for shallow frying and heat over medium.
Scoop about 1/8 cup of the mixture for each meatball and roll between your hands to form balls; wet your hands if the mixture is sticky. Arrange meatballs on a plate as you form them.
Fry the meatballs in batches in the hot oil, turning as needed, until browned on all sides, about 8–10 minutes per batch. They do not need to be fully cooked through at this stage.
Transfer browned meatballs to a plate lined with paper towels to drain excess oil, then wipe out the skillet.
Return the meatballs to the cleaned skillet and pour the 24-ounce jar of marinara sauce over them. Cover with a tight-fitting lid and simmer on low for 10–15 minutes until meatballs are cooked through.
Serve the meatballs hot, garnished with grated Parmesan or chopped parsley if desired.