Wash the chicken with water and place in pot. Cover with water and bring to a boil over high heat, skimming off bubbling foam as it forms. Add celery, carrots, onion, herbs, salt and pepper and simmer, half-covered at lower heat, for at least 45 minutes, until the chicken seems done. The chicken will come away easily from the bone. Pour soup through strainer to get a clear broth. Let cool. When broth has completely cooled, skim off the fat and save for the matzoh balls.
In a mixing bowl, mix together 4 eggs and 4 tablespoons chicken fat. Stir in the matzoh meal and salt. Add 1/4 hot water. Cover and refrigerate for at least 1 hour. Form the matzoh dough into balls the size of walnuts. Bring the salted water to a boil. Add the matzoh balls, cover, and cook for 20 minutes (don't even peek!). Bring the chicken broth to a simmer. Remove matzoh balls from hot water with a slotted spoon and add to the simmering chicken broth just a few minutes before serving.
If you are concerned if the oils or other ingredients in these recipes are suitable for Passover, seek non-dairy substitutes or ingredients that are certified kosher for Passover.