CHICKEN SOUP: Rinse the chicken with cold water and place in a large 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 to taste. Reduce heat and simmer, half-covered for at least 45 minutes, or until the chicken comes away from the bone easily.
Using tongs remove chicken to a plate. Remove skin and discard. Shred chicken with a fork. Cover and refrigerate until ready to use.
Using a slotted spoon, remove celery and carrots to a cutting board; cut into smaller pieces. Discard onion. Place veggies into a bowl; cover and refrigerate until ready to use.
Pour stock through a strainer to get a clear broth. Allow to cool. When broth has completely cooled, skim off the fat (schmaltz) and save for the matzo balls.
MATZO BALLS: In a mixing bowl, whisk together 4 eggs and 4 tablespoons chicken fat. Stir in the matzo meal and salt. Add 1/4 chicken stock. Cover and refrigerate for at least 3 hours or overnight.
Form the matzo dough into balls the size of walnuts.
Bring a pot of salted water to a boil. Add the matzo balls, cover, and cook for 20 minutes (don't peek!).
Meanwhile, in a separate pot, bring the chicken stock back to a simmer. Add chicken and veggies. Remove matzo balls from hot water with a slotted spoon and add to the simmering chicken broth just a few minutes before serving.