- Preheat oven to 275 degrees.
- Sear rabbit loins and legs in a cast iron pan over medium-high heat with olive oil.
- Braise meat in the chicken stock with remaining ingredients, covered, for 1½ hours in preheated oven. Remove pan from heat and leave it covered for 30 minutes.
- For pecan purée, smoke pecans over wood chips for 20 minutes at 215 degrees (can do this on a grill—add chips to an aluminum pan and place pecans in a pan next to it. Close cover to smoke). Transfer nuts to a pot and add chicken stock. Bring stock to a low simmer and cook for 10 minutes. Purée in a blender and season to taste with salt and pepper.
- For celery juice reduction, in a saucepot set over medium heat reduce juice by half, yielding 1 pint of juice. Add 2 cups braising liquid from the rabbit and reduce to 1 cup.
- For Brussels sprout salad, toss all ingredients together and salt and pepper to taste.
- To serve, poach rabbit legs in remaining rabbit stock to bring to temperature for 10 minutes. After 7 minutes, add the loin to cook through, about 3 minutes. Serve with celery reduction over Brussels sprout and pear salad and top with pecan purée.