Peel the carrots and cut them into thick rounds. Wash the celery and cut it into pieces.
Heat 2 tablespoons of oil in a Dutch oven. Sauté the chicken pieces for a few minutes, cover with water, add the carrots, celery, bay leaves, salt, and pepper. Bring to a boil, skim off any foam, and continue cooking over medium heat for 30 minutes.
Toward the end of the cooking time, melt the butter in a saucepan over low heat. Gradually add the flour while stirring continuously. Add 2 ladles of broth from the Dutch oven, then pour this mixture back into the blanquette. Mix well.
Clean the mushrooms and quarter them after removing their stems. Peel the pearl onions. In a skillet, sauté the mushrooms and onions over low heat in 1 tablespoon of hot oil for 5 minutes, then set aside.
In a bowl, mix the cream with the egg yolk. Just before serving, incorporate this mixture into the Dutch oven, add the onions and mushrooms, and continue cooking for 5 minutes over low heat. Adjust seasoning with salt and pepper if necessary, and sprinkle with parsley.