Real Talk: This soup is incredible due to its velvety, luxurious texture. Everyone you serve it to will be impressed. BUT you need to fully reduce the broth to 2 quarts, which should only take a half an hour; you can’t even get through an entire episode of Game of Thrones in that time. So have patience, grasshopper! And you’ll also get to brag to said guests that you learned how to temper eggs (cooking them without scrambling) thanks to this recipe. Bring chicken, carrot, leek, onion, and 4 quarts water to a simmer in a large pot; cook until chicken is cooked through, 45–60 minutes. Transfer chicken to a plate; discard carrot, leek, and onion. Let chicken sit until cool enough to handle. Continue to cook broth over medium-high heat until reduced to about 2 quarts, about 30 minutes. Season with salt. Meanwhile, shred meat from thighs and legs; set aside (reserve breasts for another use). Whisk eggs and lemon juice in a medium bowl until foamy and no streaks remain. Whisking constantly, add 1 cup hot chicken broth, a couple teaspoons at a time at first (don’t pour too much hot broth too quickly or you’ll end up with scrambled eggs), then increasing to a steady stream, until fully incorporated. Meanwhile, reduce heat to medium and add rice to broth, stirring vigorously so rice doesn’t stick to bottom of pot; cook 5 minutes. Add reserved chicken and continue to cook until chicken is heated through and rice is tender, 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 and luxurious-looking, about 5 minutes. Remove from heat and season with salt. Let sit 10 minutes to thicken up slightly. Divide soup among bowls; season with pepper and drizzle with oil. Do Ahead: Soup can be made 3 days ahead; cover and chill.