Preheat oven to 350F with rack in the middle.
Pat chicken dry and sprinkle with 1 t salt and a rounded 3/4 t pepper. Heat oil with 1 T butter in a wide 3 1/2 – 5 quart heavy ovenproof pot over medium-high heat until foam subsides, then brown chicken in 2 batches, turning once, about 10 minutes per batch. Transfer to a plate.
Meanwhile, wash leeks and pat dry.
Pour off fat from pot, then cook leeks, shallot, and 1/4 t salt in the remaining 2 T butter, covered, over medium-low heat, stirring occasionally, until leeks are pale and golden, 5-7 minutes. Add chicken, skin sides up (with juices from plate), carrots and wine and boil until liquid is reduced by half, 3-4 minutes.
Cover pot and braise chicken in oven until cooked through, 20-25 minutes. (** BAKED MINE 1 HOUR)
While chicken braises, peel potatoes, then generously cover with well-salted cold water in a 2-3 qt saucepan. Bring to a boil, then simmer until just tender, about 15 minutes. Drain in a colander, then return to saucepan. Add parsley; shake to coat.
Stir heavy cream into chicken mixture. Season with salt, pepper and lemon juice, then add potatoes.