Joint the chicken into 8 pieces or ask your butcher to joint it. Season the chicken pieces well with salt and pepper.
Heat half the oil and butter in a low casserole pan and brown the chicken pieces, in batches on all sides. Transfer to a plate.
Lower the heat, add the chopped shallots and garlic to the pan and continue to cook for 2 minutes.
Add the Semillon, stock, thyme and bay leaves, then return chicken to pan bring to a simmer, cover and leave to cook for 15 mins. Then uncover and cook for a further 15 mins.
For the sweet potato mash, boil the sweet potato chunks in salted water for 15-20 minutes or until tender. When cooked, drain in a colander and leave to steam dry for a couple of minutes.
Tip the potatoes back into the pan and mash them with a masher adding the butter and milk. Cook over a low heat till thickened stirring constantly. Season well with salt.
Meanwhile, fry the mushrooms in a little of the remaining butter and oil and season with some salt and pepper. Uncover the chicken, lift the pieces onto a large shallow serving dish and scatter over the mushrooms.
Mix the crème fraîche and egg yolks together in a small bowl. Remove the pan from the heat. Add 2 tablespoons of the pan juices to the creme fraiche mixture stirring well. Then crème fraiche liaison to the pan juices, return the pan to a low heat and stir over a very gentle heat until the sauce thickens slightly, like a thin cream consistency, but do not let it boil.
For the green salad, lightly crush the garlic clove on a board using the side of a knife. Sprinkle with a pinch of salt and mash it into a smooth paste using the blade.
Transfer to a small bowl and whisk in the red wine vinegar, Dijon mustard, and sugar.
Gradually whisk in the olive oil to emulsify, then season with ¼ teaspoon of salt.
Break the lettuce into individual leaves. Wash and dry thoroughly using a salad spinner or clean kitchen towel. Tear into bite-sized pieces and place in a large salad bowl. Just before serving, toss the lettuce leaves with about 3 tablespoons of the dressing. Serve immediately.
Pour sauce over chicken. Serve the chicken saute with the sweet potato mash and salad to the side.
