Combine the chicken broth, cream, vinegar, egg, sugar, flour, and mustard powder in a saucepan and whisk to combine. Bring the mixture to a boil over medium heat, then remove from the heat and strain through a fine-mesh sieve. Season with salt and pepper.
In a medium skillet, cook the diced bacon until slightly crisp, 6 to 8 minutes. Transfer to paper towels to drain.
Pull all of the meat off the chickens, discarding the skin and bones, and cut the meat into bite-size pieces. Place the chicken in a large bowl along with the celery, scallions, bacon, and avocados and toss gently. Add three-quarters of the dressing and toss again, adding the remainder if it seems dry.
Line a serving platter with Bibb lettuce leaves and spoon the chicken salad on top. Scatter the parsley over all.
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