Heat a slow cooker on High until hot. Stir in butter, followed by carrots, celery, onion, garlic, salt, and pepper. Cook, stirring occasionally, until golden, 6 to 8 minutes.
Stir the turkey carcass, chicken broth, thyme, sage, and bay leaf into the slow cooker. Reduce the heat to Low, cover, and cook until turkey meat is falling off the bones, about 6 hours.
While the soup is cooking, make the stuffing balls. Roll the leftover stuffing into 2-inch balls. Place bread crumbs into a shallow dish and coat balls evenly in bread crumbs.
Heat vegetable oil in a deep pan to 350 degrees F (175 degrees C).
Fry stuffing balls in the hot oil until evenly browned, about 3 minutes. Remove stuffing balls with a slotted spoon and drain on paper towels. Set aside.
Discard the turkey bones from the soup. Serve with two stuffing balls in each bowl.