In a large, heavy pot or Dutch oven filled with simmering water, cook the chicken, halved onion and bay leaf until cooked through and tender, 15 to 20 minutes. Transfer the chicken to a cutting board and cut into bite-size pieces or, using 2 forks, shred in a bowl.
Wipe out the pot, add the butter and melt over medium heat. One vegetable at a time, stir in the chopped onion, potato, carrots, celery and mushrooms. Cover the pot, lower the heat and cook until softened, about 5 minutes. Add the green beans and season with salt and pepper. Sprinkle the flour into the pot and stir for 1 minute. Stir in the wine, scraping the browned bits from the bottom of the pan, and cook until slightly reduced, about 1 minute. Stir in the chicken broth and bring to a simmer. Stir in the peas, tarragon and mustard and simmer for 5 minutes.
Add the chicken to the vegetables. Stir in the cream and heat through. Serve with the bread.