Season chicken with 1 tsp. salt and 1/2 tsp. pepper. Set aside.
Melt 1/2 cup butter in a wide 6-qt. pot or oval 9- by 14-in. dutch oven over medium heat. Add leeks and cook, stirring, 1 minute. Cover, reduce heat to medium-low, and cook leeks until meltingly soft, 10 to 15 minutes. With a slotted spoon, transfer leeks to a bowl. Increase heat under pot to medium-high. Brown chicken in a single layer, turning once, 10 to 14 minutes total. Transfer chicken to a plate.
Stir flour into fat in pot and cook 1 minute. Add wine, 1/2 cup broth, the leeks, parsley sprigs, garlic, sugar, and parsley roots. Cover and bring to a simmer over high heat; then reduce heat and simmer until parsley roots are just barely tender, 15 to 20 minutes.
Nudge chicken pieces into liquid; add more broth if needed to cover. Simmer, covered, until parsley roots are tender and chicken is cooked through, 15 minutes for white meat (transfer to a plate as done) and 20 minutes for dark. Return to pot.
Meanwhile, in a large frying pan over medium heat, cook mushrooms with remaining 2 tbsp. butter, 1/2 tsp. salt, and 1/4 tsp. pepper, stirring often, until tender and browned, 12 to 18 minutes. Add half the chopped parsley, toss to combine, and cook about 30 seconds. Remove from heat.
Whisk together egg yolks and crème fraîche in a small bowl. Set aside.
Transfer chicken and parsley roots with a slotted spoon to a warm platter. Cover with foil. Discard parsley sprigs from liquid in pot. Boil liquid over high heat until reduced to 2 cups, 4 to 7 minutes. Reduce heat to low. Spoon some of braising liquid into egg yolk mixture, stir well, then stir mixture into liquid in pot. Heat (but don't boil), stirring, until sauce is just thick enough to coat a spoon, 3 to 4 minutes. Add salt and pepper to taste. Pour into a bowl.
Rewarm mushrooms over high heat. Spoon over chicken and garnish with remaining parsley. Serve sauce on the side.
*Find parsley roots at well-stocked grocery stores and farmers' markets.