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