Cook and stir carrots, sliced onion and celery in 4 tablespoons butter in Dutch oven until onions are tender; push to side. Add chicken; cook uncovered until light brown, about 10 minutes.
Add water, 1/2 cup of the wine, bouillon, salt and 1 bouquet garni. Heat to boiling; reduce heat. Cover and simmer until thickest pieces of chicken are done, about 40 minutes.
Heat onions, 2 tablespoons butter, remaining wine and 1 bouquet garni to boiling; reduce heat. Cover and simmer until onions are tender, about 25 minutes.
Remove chicken and onions to warm platter with slotted spoon. Strain bouillon and onion liquid together; discard bouquets garni, carrots, onion and celery slices. Skim fat. Add mushrooms and lemon juice and heat to boiling; reduce heat. Simmer uncovered until reduced to 2 1/2 cups.
Mix egg yolks and whipping cream. Beat 1 cup hot broth by tablespoonsful into whipping cream mixture. Beat in remaining broth. Heat to boiling, stirring constantly. Boil and stir 1 minute: pour over chicken and onions. Sprinkle with parsley.
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