In a large Dutch oven or stockpot (at least 5-quarts), combine the chicken thighs, cold water, quartered onion (with skin), whole peppercorns, and kosher salt. Cover with a tight-fitting lid and bring the mixture to a boil over high heat. Once boiling, reduce the heat to maintain a gentle simmer. Cook for 45 minutes to 1 hour, skimming and discarding any foam that rises to the surface with a slotted spoon.

Once the chicken has cooked thoroughly, use tongs to transfer the thighs to a clean cutting board. Strain the broth through a fine mesh sieve into a large heatproof bowl, discarding the solids. Skim excess fat from the broth if desired. Reserve 2 cups of the strained broth in a separate measuring cup for tempering the eggs later. Return the remaining broth to the Dutch oven and keep it warm over low heat. Remove and discard the chicken skin and bones. Shred the meat into bite-sized pieces and set aside.

Bring the strained broth in the Dutch oven back to a gentle boil over medium-high heat. Add the dried orzo and cook for 7 to 9 minutes, stirring occasionally, until the pasta is al dente. Reduce heat to low and stir in the shredded chicken.

In a medium mixing bowl, whisk the eggs vigorously for approximately 2 minutes, or until light and frothy. Slowly add the lemon juice while continuing to whisk. Gradually pour the reserved 2 cups of warm broth into the egg-lemon mixture, whisking constantly to temper the eggs and prevent curdling.

Slowly pour the tempered egg-lemon mixture into the Dutch oven, stirring gently to incorporate. Cook the soup over low heat for 3 to 5 minutes, stirring frequently. Do not allow the soup to boil, as this can cause the eggs to curdle. The soup should thicken slightly to a creamy consistency.

Ladle the soup into bowls and garnish with lemon rounds, fresh dill or oregano, and a light dusting of freshly ground black pepper if desired. Serve immediately.