In a large stockpot or Dutch oven, heat the oil over medium-high heat. Season the chicken generously with salt and pepper. Add to the pot in an even layer and sear until a nice crust has formed on both sides, about 3 minutes per side. Transfer to a plate.

Add the fennel and carrots to the pot. Sweat the veggies over medium heat, stirring occasionally and scraping up any brown bits that may have formed from the chicken, until tender, about 5 minutes. Return the chicken to the pot along with the broth and 1 teaspoon of salt. Bring to a boil, lower the heat, and gently simmer until the carrots are soft and the chicken is tender enough to pull apart with a fork, about 15 minutes.

Remove the chicken from the pot again and transfer to a medium-size bowl. Shred with a fork into bite-size pieces.

Meanwhile, add the zucchini noodles to the broth and simmer over low heat until tender, about 3 minutes. Remove the pot from the heat and add the lemon juice, shredded chicken, and dill.

In a medium-size bowl or 4-cup measuring cup, beat the eggs.

Temper the egg by slowly whisking in 1?4 cup of the hot broth into them, adding a very small amount at a time so as not to scramble the egg. Repeat with another 3/4 cup of broth. Once the egg mixture is warm to the touch and the broth is fully incorporated, stir the mixture into the soup. If your egg “breaks” into stringy bits, don’t freak out. It will still taste delicious, but will look a little funky—like Chinese egg drop soup. Pretend that’s what you were going for and, either way, serve immediately.