Place the butter in a large pot over low heat and melt. Add the carrots, celery, onion, zucchini and garlic to the pot. Stir and then put the lid on the pot and let sweat for 30 minutes.
Add the chicken breasts, broth, salt and pepper and bring to a boil over high heat. Reduce heat to medium-low and continue to simmer until chicken is cooked through, about 8 minutes. Using a pair of tongs, remove the chicken and place on a cutting board. Cut the chicken into bite-size pieces and then add back to the broth mixture.
In a small bowl, whisk together the egg yolks and lemon juice. Slowly add 1/2 cup of the hot liquid to the egg yolks while whisking constantly. Then, slowly add the yolk mixture to the soup while whisking constantly.
Add the parsley, season with additional sea salt and pepper if needed, and serve.
