Preheat an outdoor grill for medium-high heat, and lightly oil the grate. Stir the cranberry sauce, horseradish, lemon juice, and cumin together in a bowl. Season to taste with salt and pepper, and set aside.
Mix the egg, celery, and onion in a bowl with your hand until the egg is smooth. Add the turkey, thyme, poultry seasoning, parsley, and bread crumbs; mix until evenly combined. Form the turkey mixture into 8 patties.
Cook the turkey burgers on the preheated grill until no longer pink in the center and the juices run clear, about 4 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Place onto the hamburger buns, and top with cranberry sauce to serve.
