Season the chicken pieces with salt and pepper. In a large, heavy-bottomed pot, heat the olive oil or butter over medium-high heat and brown the chicken pieces. Remove and set aside.
Add the white wine to the pot to deglaze, scraping up any brown bits. Return the chicken to the pot and add the chicken stock, bay leaf, thyme, onion, carrots, and celery. Bring to a simmer, cover, and cook gently until the chicken is tender.
In a separate pan, melt the butter, then stir in the flour to make a roux. Cook gently without coloring, for about 2 minutes.
Sauté the pearl onions (and lardons/pancetta if using) until golden and cooked through. Sauté the mushrooms until they release their moisture and become golden.
Remove the cooked chicken and vegetables from the pot. Strain the cooking liquid and return it to the pot. Whisk in the roux and bring to a simmer to thicken. Then, blend the heavy cream and egg yolk in a bowl and slowly incorporate some of the hot cooking liquid to temper it. Gradually whisk this mixture back into the pot. Add the lemon juice, and adjust seasoning.
Return the chicken and all vegetables to the pot, warming everything through gently. Sprinkle with chopped parsley before serving.