How to Make It
Preheat the oven to 400°. Place the chicken in a small roasting pan. Rub 1 tablespoon of the butter over the chicken and season with salt and pepper. Roast for about 1 hour and 20 minutes, rotating the pan once, until the skin is browned and crisp and the chicken is cooked.
Meanwhile, in a medium saucepan, melt 1 tablespoon of the butter. Add the wing tips, neck, gizzard, onion and carrot, season with salt and pepper and cook over moderate heat until browned all over, about 10 minutes. Add the wine and simmer over moderately high heat until reduced to 2 tablespoons, about 4 minutes. Add the stock and bring to a boil. Simmer over low heat for 20 minutes.
In a large skillet, bring 1/2 inch of water to a boil. Add the sugar snaps and cook over high heat until bright green but still crisp, about 2 minutes. Transfer to a plate and set aside.
Wipe out the skillet and heat 2 tablespoons of the olive oil. Add the white mushrooms, season with salt and pepper and cook over high heat for 1 minute. Stir and cook over moderate heat until any exuded juices evaporate and the mushrooms start to brown, about 7 minutes. Add the remaining 1/2 tablespoon of olive oil to the skillet. Add the oyster mushrooms, season with salt and pepper and cook over moderately high heat until browned on the bottom, about 4 minutes. Stir and cook until browned all over, about 3 minutes longer. Add the shallot and cook, stirring, until softened, about 2 minutes. Remove the mushroom mixture from the heat.
When the chicken is done, tilt it to release the juices from the cavity into the pan. Transfer the chicken to a carving board and keep warm. Pour the pan juices into the saucepan with the stock, scraping in any browned bits. Add the tarragon sprigs and simmer the stock over moderate heat for 5 minutes. Strain the jus through a sieve into a small saucepan and skim the fat from the surface. Boil over high heat until reduced to 3/4 cup, about 4 minutes. Stir in half of the tarragon leaves and season with salt and pepper.
Reheat the mushrooms over moderately high heat. Add the remaining 1/2 tablespoon of butter and the blanched sugar snaps and cook, stirring, until heated through. Season with salt and pepper and stir in the remaining tarragon leaves.
Carve the chicken and arrange on 4 plates. Spoon the mushrooms and sugar snaps alongside and pass the jus at the table.