Heat 1 tablespoon oil in Dutch oven over medium-high heat until just smoking. Pat chicken dry with paper towels and season with salt and pepper. Brown chicken lightly on both sides, about 5 minutes; transfer to plate.

Add remaining 1 tablespoon oil, carrots, fennel, and onion to fat left in pot and cook over medium heat until vegetables are softened, about 5 minutes. Stir in thyme and cook until fragrant, about 30 seconds. Stir in broth and lemon zest, scraping up any browned bits.

Add browned chicken, cover, and simmer gently until it registers 160˚F, about 10 minutes.Transfer chicken to cutting board and shred into bite-size pieces.

Meanwhile, remove lemon zest and return soup to simmer. Stir in shredded chicken and rice and cook until heated through, about 1 minute.

Turn heat to low.

Whisk egg yolks and lemon juice together in a medium bowl.Stir 3 tablespoons hot broth into yolk mixture to temper (so it doesn't curdle)then whisk yolk mixture into soup. Continue to cook soup gently, whisking constantly,, until it thickens slightly, about 1 minute. Promptly remove from heat. Stir in parsley and season with salt and pepper to taste. Serve immediately.