The next day, drain the capon pieces and pat dry with a cloth.
Heat the olive oil in a large skillet. When it begins to smoke, brown the capon pieces for 5 minutes on each side.
Drain the capon pieces and transfer to a preheated Dutch oven containing 3 tablespoons of heated butter. Add the drained diced carrot and onion from the marinade ; continue sautéing, sprinkling the sugar over top, then the flour, and cook uncovered for 5 minutes, stirring with a wooden spatula.
Place the quartered pig's feet on top of the mixture. Pour on the Armagnac, flambé, and add the wine from the marinade with the bouquet garni, garlic and chicken stock.
Bring to a boil, cover and cook in a low oven at a bare simmer for 1 3/4 hours. A half hour before the end of the cooking time, add the pearl onions and mushrooms, sautéed until golden brown, to the pot.
Finishing and serving
Once everything is cooked, remove the capon pieces, pig's feet, pearl onions, mushrooms and bouquet garni with a skimmer. Remove the toothpicks and keep everything hot.
Pour the cooking liquid into a blender with half the diced carrot and onion and the orange peel. Purée to make a smooth sauce.
Return the capon, pig's feet and remaining diced vegetables to the Dutch oven (or a more elegant serving dish, if you wish) .
Strain the sauce through a sieve and pour over the capon. Sprinkle the pearl onions, mushrooms and coarsely-chopped parsley over top.