Heat a large skillet over medium heat. Melt the butter and add the chopped onions, celery, and apple, then season with salt and pepper and cook for 3 to 4 minutes under a loose foil tent. Stir in the bread crumbs and transfer to a bowl to cool a bit. Add the meat, poultry seasoning, salt, pepper, parsley, mustard, and the egg and mix. Form 4 burgers. Wipe out the skillet and heat a liberal drizzle of EVOO in it over medium-high heat. Cook the patties for 11 to 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.
Mix together in a small bowl the cranberry sauce, sour cream, and chives.