Pour enough vegetable oil into a large, deep pot so that it reaches a depth of 1 inch. Warm oil over medium heat until a deep-fry thermometer registers 325°F. Line a baking sheet with parchment or foil, and line a plate with paper towels
While vegetable oil is heating, whisk together egg and milk in a shallow bowl. Place bread crumbs in a separate shallow bowl. Working in batches, dip ravioli into egg mixture, allowing excess to drip back into bowl, then coat with bread crumbs. Place coated ravioli on baking sheet. Discard remaining bread crumbs.
Fry ravioli in batches, turning occasionally, until golden brown, about 3 minutes total. Be careful not to crowd ravioli in pan, and make sure oil comes back to 325°F before adding another batch of ravioli. Transfer fried ravioli to lined plate to drain. Sprinkle fried ravioli with grated Parmesan.
Warm marinara sauce in a pan over medium-low heat or in a microwave. Spoon sauce into 4 small bowls. Serve fried ravioli with warmed marinara on the side.