Prepare the Broth: In a 5-quart or larger Dutch oven, combine the chicken thighs, cold water, quartered onion, black peppercorns, and kosher salt. Cover and bring to a boil over high heat. Reduce heat to maintain a gentle simmer. Cook until the chicken is fully tender and cooked through, 45 minutes to 1 hour. If foam forms on the surface, skim and discard using a slotted spoon.
Strain and Shred: Using tongs, transfer the cooked chicken to a cutting board. Strain the broth through a fine-mesh sieve into a large heatproof bowl; discard the solids. If desired, skim excess fat from the broth using a spoon or fat separator. Reserve 2 cups of the broth in a separate measuring cup. Return the remaining broth to the Dutch oven and keep warm over low heat.
Shred the Chicken: Once the chicken is cool enough to handle, remove and discard the skin and bones. Shred the meat into bite-sized pieces using your hands or two forks. Set aside.
Cook the Orzo: Bring the strained broth in the Dutch oven back to a boil over medium-high heat. Add the orzo and cook, stirring occasionally, until al dente, 7-9 minutes. Stir in the shredded chicken and reduce heat to low.
Prepare the Avgolemono Sauce: In a medium bowl, vigorously whisk the eggs until lightened in color and frothy, about 2 minutes. While continuously whisking, slowly pour in the lemon juice. Gradually drizzle the reserved 2 cups of warm broth into the egg-lemon mixture, whisking constantly to temper the eggs and prevent curdling.
Thicken the Soup: Pour the tempered egg-lemon mixture into the Dutch oven with the chicken and orzo. Stir gently to combine. Cook over low heat, stirring frequently, until the soup thickens slightly, 3-5 minutes. Do not allow the soup to boil.
Serve: Ladle the soup into bowls. Garnish with lemon slices, fresh herbs, and a sprinkle of black pepper. Serve immediately.