Heat 2 tablespoons of oil in a skillet over medium-high heat. Add the celery, onion, carrot and parsley, and cook until softened, about 5 minutes. Let cool slightly.
Heat a 1/4 of an inch of oil in a large skillet with high sides over medium-high heat.
In a mixing bowl, shred the leftover turkey using two forks. Add the mashed potatoes, 1/2 cup of the breadcrumbs, poultry seasoning, nutmeg, salt, pepper, egg and cooled veggies. Mix to combine and form 10 patties about an inch thick.
Dredge the patties in the remaining breadcrumbs and brown in two batches in a single layer in the hot oil for 2 minutes on each side or until deep golden brown. Drain on paper towels and transfer to a serving plate. Serve with cranberry sauce and gravy alongside.