Preheat oven to 350°F.

In a small pot, steam potatoes over high heat until just tender when pierced with a fork, about 25 minutes. Set aside.

Pat chicken dry and sprinkle with sea salt and freshly ground pepper on both sides.

Heat oil and butter in a wide 5-quart heavy ovenproof pot (with lid) over medium-high heat until foam subsides, then brown chicken in 2 batches, turning once, about 10 minutes total per batch. Make sure the skin is browned. Transfer to a plate.

While chicken is cooking, chop leeks and place in a bowl of cold water to remove dirt. Let sit for 10 minutes. Transfer leeks to a colander by hand or spoon. Rinse leeks under cold water in colander. Pat dry.

Pour off all but 2 tablespoons of fat from pot. Cook leeks, shallot, garlic, carrots and mushrooms, covered, over medium-low heat, stirring occasionally, until leeks are pale golden, 8 to 10 minutes.

Add chicken, skin sides up and wine and boil until liquid is reduced by half, 5 to 7 minutes.

Add potatoes, rosemary and thyme and stir. Cover pot and braise chicken in oven until cooked through (meat should be falling off the bones), 45 to 60 minutes.

Remove pot from oven. Transfer chicken to a plate – don’t worry if some pieces fall into the pan.

Stir crème fraîche into pot. Then stir in parsley and lemon juice. Season with sea salt and freshly ground pepper. Return chicken to pot.

Serve warm.