Peel and dice the carrots and parsnips into uniform 1/4-inch pieces so they cook evenly. Dice the celery stalks, chop the onion, and thinly slice the leek (using only the white and light green parts). Mince the garlic cloves and finely chop the fresh parsley. Having everything prepped and ready before you start cooking ensures the soup comes together smoothly without interruption.
Heat the olive oil in a large pot over medium heat until shimmering. Add the diced carrots, parsnips, celery, onion, and sliced leek, and sauté for 4-5 minutes, stirring occasionally, until the vegetables begin to soften and release their flavors. This initial cooking builds a flavorful base for the broth—I like to let them cook undisturbed for the first couple of minutes to encourage light browning on the cut surfaces.
Add the minced garlic, salt, and black pepper to the pot and stir constantly for about 1 minute. This brief cooking allows the garlic to become fragrant without burning, which would make it bitter. The aromatic vegetables from Step 2 have softened enough to absorb these flavors.
Pour in the chicken broth, then nestle the chicken breasts into the aromatic base from Step 3. Add the fresh thyme sprigs, tarragon sprigs, and bay leaf. Bring the liquid to a boil over medium-high heat, then immediately reduce the heat to low and cover the pot with a lid. The chicken will poach gently in the flavorful broth, becoming tender and infusing the soup with its flavor.
Simmer the covered pot for approximately 15 minutes until the chicken is completely cooked through (the internal temperature should reach 165°F). Remove the chicken breasts to a cutting board and let them cool for a minute or two until they're cool enough to handle. Using two forks, shred the chicken into bite-sized pieces by pulling the meat apart along the grain. Return the shredded chicken to the pot and simmer for 1-2 minutes to warm it through and allow it to absorb more of the broth's flavor.
Remove and discard the fresh thyme sprigs, tarragon sprigs, and bay leaf from the pot. Stir in the dried dill and the finely chopped fresh parsley. I find that adding the fresh herbs at the very end preserves their bright color and flavor—the parsley especially adds a fresh note that makes the soup taste lighter and more vibrant. Taste the soup and adjust seasoning with additional salt and pepper if needed.
Ladle the chicken soup into bowls and serve hot. The soup is now ready to enjoy.