In a large pot, place the whole cleaned chicken and pour in the chicken broth until the chicken is fully submerged. Heat over medium-high until the broth reaches a vigorous boil.
Once the broth is boiling, reduce the heat to medium-low, and skim off any foam that rises to the surface with a ladle to ensure a clear broth.
Add the chopped onion, diced carrots, and diced celery to the pot. Cover and let simmer gently for about 1 hour, or until the chicken is cooked through and tender, easily pulling away from the bone.
Carefully remove the chicken from the pot with tongs and set it on a plate to cool slightly. Strain the broth through a fine-mesh sieve to remove the vegetables and return the clear broth back into the pot.
In a mixing bowl, vigorously whisk together the eggs, fresh lemon juice, and lemon zest until smooth and well combined.
Once the chicken is cool enough to handle, shred the meat into bite-sized pieces, discarding the bones and skin.
Bring the clear broth back to a gentle simmer over low heat, then add the orzo pasta. Cook according to the package instructions, usually about 8-10 minutes, until al dente.
To temper the egg mixture, slowly add a ladle of the hot broth into the egg mixture while whisking continuously. This will prevent the eggs from curdling. Repeat this process 2-3 times to gradually warm the eggs without cooking them.
Once tempered, slowly stir the egg mixture back into the pot of simmering soup while stirring constantly to create a creamy texture without curdling.
Gently fold in the shredded chicken and season the soup with salt and freshly ground pepper to taste.
Serve the soup hot in deep bowls, garnished with fresh dill or parsley for a burst of color and flavor.
Notes
For an inviting presentation, serve with lemon wedges on the side and a sprig of dill or parsley on top of each bowl.