Preheat the oven to 275°C.
Sauté the garlic, celery, onions, potatoes, carrots, and half the thyme in olive oil for 10 minutes.
Mix the chopped fat, chicken livers, and wine to create the stuffing.
Stuff the chickens, stitch the openings closed, and rub the skin with olive oil, salt, and pepper.
Sprinkle with remaining thyme, wrap each chicken in five layers of foil, and bake for 30 minutes.
Open the foil carefully and reserve the juices in a bowl.
Halve the chickens with shears and remove the stuffing.
Serve the chicken over the stuffing, drizzled with the reserved juices and extra oil.