To begin: Cut the breasts from the chickens and refrigerate them until you are ready to use them.
For the Broth: Put the rest of the 2 chickens into a large pot. Cover them with cold water. Bring the water to a simmer over medium-high heat, but do not let it boil. Skim off any fat and scum that rises to the surface. Once the water is simmering add the carrot, celery, onion, garlic, bay leaves, peppercorns, and white wine. Simmer slowly for 3 to 4 hours. Remove the chicken from the pot, set it aside to cool, cover, and refrigerate it. Strain the broth through a fine mesh strainer and refrigerate until ready for use. The broth can also be frozen.
For the Poached Breasts: Season the breasts with salt and pepper. Place each breast into a small resealable plastic bag. Divide the butter, thyme, wine, garlic, and chicken broth evenly into the bags. Press out all of the air and seal the bags tightly. Keep refrigerated until ready to cook.
Bring a large pot of water to a boil over high heat, then reduce the heat and bring it to a simmer. Add the bags to the water and submerge them; don't let the plastic catch the rim of the pan or they will melt. Poach them, stirring occasionally, until the chicken is firm and cooked through, about 10 minutes. Remove the bags from the water and serve the chicken immediately, using the liquid in the bags as the sauce.
For the Salad: Pick the meat from the cooled chicken, discarding the skin and bones. Cut the meat into small pieces and put them into a bowl. Add the celery, onion, celery leaves, parsley, mayonnaise, mustard, vinegar, and salt and pepper, to taste. Mix until everything is evenly blended. Serve with salad leaves or make a sandwich.
The chicken breasts we used weigh about 4 1/2 ounces each. If you use larger breasts, the cooking time will be longer.