Meanwhile, shred thighs, legs, and breast meat.
Whisk eggs and lemon juice in a medium bowl until foamy. Whisking constantly, add a couple teaspoons of hot broth to the egg mixture to temper and then add 1 cup until fully incorporated.
Meanwhile, reduce heat to medium and add rice to broth, stirring vigorously ; cook 6-8 minutes. Add chicken and continue to cook about 5 minutes more.
Whisking vigorously, slowly drizzle in tempered egg mixture, a couple teaspoons at a time at first, then increasing to a steady stream, until fully incorporated (it will cook instantly).
Reduce heat to medium-low and cook until soup is slightly thickened and velvety , about 5 minutes. Remove from heat and season with salt. Let sit 10 minutes to thicken up slightly.
Divide soup among bowls; season with fresh dill, pepper and drizzle with oil.
