Place the chicken bones and carcass into a large stockpot, then cover with cold water. Bring to a gentle simmer over medium heat, and let cook for 2 hours, skimming any foam that rises to the surface.
While the broth simmers, chop the carrots, celery, and onion into bite-sized pieces and set aside.
Once the broth has simmered for 2 hours, remove the chicken bones and carcass from the pot. Strain the broth through a fine mesh strainer into another pot, discarding solids.
Return the strained broth to the stove, and bring it back to a gentle simmer. Add the chopped vegetables and cook for 20-30 minutes, until tender and fragrant.
Stir in the shredded cooked chicken, chopped parsley, and dill. Taste and season with salt and pepper as needed.
Finish the soup by stirring in a splash of lemon juice for brightness, then cook for another 2 minutes to meld flavors.
Serve the hot chicken soup in bowls, garnished with extra herbs if desired, and enjoy its comforting, clear, and fragrant goodness.