Preheat the oven to 350°. Set the head of garlic on a double layer of foil, cut side up. Drizzle with olive oil, then wrap in the foil. Roast the garlic until very soft, about 1 hour and 30 minutes. Let cool, then peel, keeping the cloves intact.
Meanwhile, in a heatproof bowl, cover the porcini with the boiling water and let stand until softened, about 15 minutes. Rinse the porcini and coarsely chop them; reserve the soaking liquid.
In a large skillet, heat 1 tablespoon of the olive oil. Season the chicken with salt and pepper and spread in a single layer in the skillet. Cook over high heat until browned on the bottom, about 4 minutes. Transfer to a bowl.
Add 1 tablespoon of the olive oil to the skillet. Add the assorted mushrooms and season with salt and pepper. Cover and cook over moderate heat, stirring a few times, until browned and their liquid has evaporated, about 5 minutes. Transfer the mushrooms to a plate.
In the skillet, melt the 1 tablespoon of butter in the remaining 1 tablespoon of olive oil. Add the shallots and cook over moderate heat, stirring, until softened, about 3 minutes. Add the red wine and boil over moderately high heat until reduced by half, about 2 minutes. Pour in the reserved porcini soaking liquid, stopping before you reach the grit at the bottom. Add the bone broth, tomatoes, mushrooms, porcini, roasted garlic and chicken and bring to a simmer. Remove from the heat. Add the tarragon and season with salt and pepper. Swirl in the 2 tablespoons of chilled butter, 1 tablespoon at a time.