Wash the rabbit parts and pat dry. Bring 1/8 litre water and the spice mixture to the boil. Leave to cool and mix with the buttermilk. Pour over the rabbit and marinate covered for 24 hours.
Turn now and then. Clean and wash the Brussels sprouts and cook in boiling salted water for five minutes. Remove with a skimmer. Remove the rabbit from the marinade and drain well. Peel onions and cut into slices.
Heat clarified butter in a roaster and fry the rabbit parts in it. Season with salt, pepper and thyme. Add onions, fry briefly and deglaze with broth. Braise in the preheated oven (electric cooker: 200°C/ gas: level 3) for one hour.
Wash and quarter the apples, remove the core and cut the quarters into pieces. Halfway through cooking, add the sprouts and apples to the rabbit and braise. Drain the cranberries on a sieve and collect the juice.
Season the braised rabbit with salt, pepper and cranberry juice. Sprinkle with cranberries and garnish with fresh herbs as desired.