Both the vinaigrette and the poaching liquid for the chicken are prepared with fresh tarragon. (When buying this herb, look for true French tarragon, which has a delicate anise flavor, rather than the coarser Russian variety.)
Make a bouquet garni: Tie tarragon stems, thyme, bay leaves, and peppercorns in a small square of cheesecloth. Place bouquet garni, sherry, celery, onion, and 5 cups water in a medium pot, and bring to a boil over high heat. Reduce heat, and let simmer for 5 minutes.
Season chicken breasts on both sides with salt and pepper. Add to the gently simmering liquid, and cook until firm and juices run clear when pierced with a two-pronged fork, 10 to 12 minutes.
Remove chicken from poaching liquid, and arrange on serving plates. Drizzle with lemon vinaigrette.