Simmer chicken, including giblets and neck, in water until thickest pieces are fork-tender, about 2 1/2 hours.
Remove chicken; cool; spoon off fat from broth and reserve.
This much can be done the day or morning before.
About 40 minutes before mealtime, measure 1/4 cup fat and 2 cups broth (reserve remaining fat and broth for future cooking).
Cook onion in the 1/4 cup chicken fat in skillet until tender but not brown.
Add 1 cup of the broth, salt, chili powder, and pepper.
Simmer, uncovered, for 5 minutes.
Add lemon juice, wine, tarragon, and ginger and the second cup of broth.
Place chicken and eggs in the sauce.
Cover and simmer about 30 minutes or until chicken and eggs have absorbed some of the color from the sauce.
Serve hot.