Put half the onion and half the carrots on the bottom of a baking dish just large enough to hold the rabbit pieces in one layer. Lay the rabbit on top, then press the remaining onions and carrots lightly into the meat. Pour the cider over the rabbit and press the thyme sprig down lightly into the meat. Cover the bowl and marinate the rabbit for about 24 hours in the refrigerator, turning 2 or 3 times.
Preheat oven to 350 degrees F.
Remove the rabbit from the marinade. Strain the marinade, reserving the cider and vegetables separately. Pat the rabbit dry.
Heat the oil in a Dutch oven or another heavy-bottomed ovenproof pot over high heat. Fry the rabbit, turning frequently, for 6 to 8 minutes, or until browned on all sides. Remove the rabbit from the Dutch oven and set aside. Reduce the heat to low, add the reserved vegetables, and cook, stirring frequently, for 10 to 12 minutes, or until the vegetables are soft and starting to brown. Sprinkle flour over the vegetables and stir well.
Slowly pour in the reserved cider and then the stock, stirring well. Season to taste with salt and pepper, then return the rabbit to the Dutch oven, turning it over in the liquid so that it is well coated. Cover the pot and braise the rabbit in the oven for 30 minutes. Lower the temperature to 300 degrees F, turn the rabbit pieces over in liquid and cook for another 1 1/2 hours, turning the rabbit every 30 minutes.