Place 6 bone-in, skin-on chicken thighs, 8 cups cold water, 1 large unpeeled quartered yellow onion, 2 tablespoons whole black peppercorns, and 1 tablespoon kosher salt in a 5-quart or larger Dutch oven. Cover and bring to a boil over high heat. Reduce heat to maintain a simmer and simmer until the chicken is cooked through, about 45 minutes to 1 hour. If any white foam forms, use a slotted spoon to skim off and discard.
Transfer the chicken to a cutting board. Strain the stock through a fine-mesh strainer set over a large heatproof bowl and discard the solids. If there is an abundance of fat rendered from the chicken, skim it off it with a spoon or use a fat separator. Reserve 2 cups of the stock in a measuring cup. Return the remaining stock to the Dutch oven and keep warm over low heat.
When the chicken is cool enough to handle, use your hands to shred the meat into bite-sized pieces; set aside. Discard the skin and bones.
Bring the stock back to a boil over medium-high heat. Add ½ cup dried orzo and cook until al dente, 7 to 9 minutes. Stir in the reserved shredded chicken. Reduce the heat to low.
Place 4 large eggs in a medium bowl and whisk until lightened in color and frothy, about 2 minutes. While whisking, slowly pour in ¼ cup freshly squeezed lemon juice. While still whisking, temper in the eggs by slowly drizzling the reserved 2 cups of warm stock into the egg-lemon mixture. This warms the eggs just enough so that they do not curdle when added to the hot soup.
Add the avgolemono back into the pot with the chicken and orzo and stir to combine. Cook until the soup thickens slightly, 3 to 5 minutes, but do not let it come to a boil.
Pour the soup into serving bowls and serve with lemon slices, fresh chopped dill or oregano, and freshly ground black pepper.