Cover the porcini with water in a small saucepan, bring to a boil, turn off the heat, and allow to soak.
Wash the breasts and pat dry. Remove the small rib bones with a knife and cut the breasts into small serving pieces. (A large breast half might be cut into three pieces.) Season each piece on both sides with salt, pepper, and dried herbs. Roughly chop the pancetta. Finely chop the celery, carrot, and onion. Chop the tarragon leaves, discarding the stems. Drain the tomatoes and crush by hand. Peel the garlic and chop very coarsely
Heat 2 tablespoons of butter and 2 tablespoons of olive oil in a large sautépan. Brown the chicken on both sides over medium-high heat and remove to a Dutch oven or other pot.
Sauté the pancetta in the same pan for two minutes. Add the celery, carrot, and half the onion with some more dried herbs and more butter and oil if necessary. Cook until soft and translucent. Transfer to the pot with the chicken. Sauté the garlic until it begins to brown. Add to the chicken and vegetables along with the tomatoes, wine, and 1/3 of the tarragon. Chop the porcini and add them to the pot along with their liquid. Bring to a boil, cover, and reduce the heat to a simmer.
Wash, dry, and quarter the mushrooms. Heat 2 tablespoons of butter and 2 tablespoons of olive oil in the pan and cook the mushrooms, seasoned with salt and pepper, with the remaining onion until most of the liquid has evaporated. Add the Madeira and half the remaining tarragon and continue to cook for several minutes.
When the chicken has simmered for thirty minutes, mix in the mushrooms and continue to cook for fifteen minutes more or until the chicken is tender. Meanwhile, prepare a beurre manié by kneading 2 tablespoons of butter with 2 tablespoons of flour. When the chicken is done, while continuing to simmer, gradually stir the beurre manié into the sauce, a small piece at a time, until the sauce has somewhat thickened. Cover with the remaining tarragon and serve.