Place the morel mushrooms in a bowl, cover them with hot water, and leave them to soak for 30 minutes. Drain them and cut them in half. Pour the Madeira into a saucepan and reduce it by half. Add the morel mushrooms and half a chicken stock cube. Cover with water, and leave to cook uncovered for 40 minutes over medium heat.
Salt the flesh side of the chicken pieces. Cut off the stalks of the button mushrooms. Cut the tops into thin strips.
Peel the shallots and cut them into thin strips. Rinse and dry the tarragon.
Pour 1cup (250ml) of water into a saucepan with the Noilly Prat and the white wine. Add the tarragon, the shallots, the mushrooms, and two stock cubes.
Heat over a high flame.
Put the chicken pieces into the pan and let them cook for 12 minutes, uncovered. At the end of the cooking time, take out the pieces of white meat. Leave the dark meat to cook for another 13 minutes.
Beat the butter to soften* it. Add the flour and mix well to make a beurre manié.
Take the pieces of dark meat out of the pan. Take out the tarragon. Reduce the cooking liquid until it is dry*. When it “squeaks,” there will just be fat left and the gravy will be completely reduced.
Then add the beurre manié.
Add the crème fraîche right away, and cook for 5 minutes, stirring constantly. Put the pieces of chicken back into the pan. Turn them over several times in the sauce, and let them warm up again.
Pour the blanquette into a warmed serving dish. Drain the morel mushrooms and add them, together with alittle freshly chopped tarragon. Serve at once.