Heat a large skillet over medium heat. Melt the butter and add the chopped onion, celery and apple, then season with salt and pepper and cook for 3-4 minutes under a loose foil tent. Stir in the breadcrumbs and transfer to a bowl to cool a bit. Add the meat, poultry seasoning, salt, pepper, parsley, mustard and the egg and mix. Form four burgers. Wipe out the skillet and heat a liberal drizzle of EVOO in it over medium-high heat. Cook the patties for 11-12 minutes, until cooked through. Melt the cheddar over the tops of the burgers during the last minute or two of cooking.
While the burgers cook, toast the split English muffins under the broiler or in the toaster.
In a small bowl, mix together the cranberry sauce, sour cream and chives.
Serve the burgers on the English muffins with red onion, lettuce and the cranberry sauce.