Tie the parsley, tarragon, and thyme sprigs and bay leaf together with kitchen twine and put in a large saucepan with the onion, carrot, celery, lemon zest, and chicken breasts. Cover with the broth, bring just to a boil over high heat, skim off any foam that comes to the surface. Adjust the heat to very low and cover. Cook the chicken until firm to the touch, about 20 minutes. Remove the chicken to a platter, when cool enough to handle and pull into large strips; discard the bones.
When ready to serve, add the asparagus, peas, and mushrooms to the broth. Cook until the vegetables are just tender, about 3 to 5 minutes, and remove herb bundle. Return chicken to the broth and warm through.
Divide chicken between 4 large soup bowls and ladle some broth and vegetables into each bowl. Garnish each soup with the chopped parsley and tarragon. Serve.
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