1. Tie parsley, tarragon, and thyme sprigs and bay leaf together with kitchen twine and place in a large saucepan with onion, carrots, celery, lemon zest, and chicken breasts. Pour in broth and bring to a boil over high heat. Immediately reduce heat to low and skim surface. Cover and cook until chicken is firm to touch, about 20 minutes. Remove from heat and transfer chicken to a platter. When chicken is cool enough to handle, pull meat in large strips from bones; discard bones.
2. Add asparagus, peas, and mushrooms to broth. Cook until vegetables are tender, about 3 to 5 minutes. Remove herb bundle, return chicken to broth and cook until heated through, 2 to 3 more minutes. Pour soup into 4 large bowls and garnish with chopped parsley and tarragon.
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