Remove chicken bones; break them up and combine in a pot with the celery, carrots, and water. Simmer, partly covered, for ¾-1 hour. Uncover; boil to reduce stock to 1 cup. Meanwhile, discard chicken skin and loose fat. Slice meat in julienne strips; reserve.
Melt butter in wide, heavy skillet; add shallots and stir over moderately low heat until slightly soften