Heat a large skillet over medium heat. Add the butter to melt, then add the apple, onion and celery; season with salt and pepper. Cover and cook until softened, 3 to 4 minutes. Stir in the breadcrumbs, then transfer to a bowl to cool; reserve the skillet. Mix in the turkey, egg, parsley, mustard and poultry seasoning; season with salt and pepper. Form into four 4-inch patties.
Wipe out the reserved skillet and add a liberal drizzle of EVOO. Warm over medium-high heat, add the patties and cook, turning once, until cooked through, 8 to 10 minutes. Top with the cheddar during the last minute.
Meanwhile, in a small bowl, mix the cranberry sauce, sour cream and chives.
Serve the patties on the English muffins with the onion, lettuce and cranberry-sour cream sauce.