Place the rabbit n a large mixing bowl, then pour over the wine, olive oil, garlic, carrots, cloves, bay leaves, celery, onions, thyme, salt and pepper. Cover and refrigerate 8 hours or overnight to marinate.
Remove the rabbit from the marinade and set aside. Transfer the marinade (along with the solids) to a Dutch oven and bring to a boil. Cook for 10 minutes, then reduce the heat and add the potatoes. Cook 10 minutes more.
Add the rabbit, cover and cook until the rabbit is cooked through, 15 to 20 minutes.
Transfer all the solids from the pan to a large serving platter and cover to keep warm. Over high heat, reduce the liquid in the Dutch oven until reduced by half, about 10 minutes.
Add the butter and whisk until the sauce becomes nice and shiny, then add the mushrooms and cook 2 minutes more. Spoon the vegetables onto the platter with the rabbit, then pour the red wine sauce on top. Garnish with the parsley and serve immediately with the baguette.
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