Cut carrots into 1/2-inch rounds, slice celery into 1/4-inch crescents, and slice leeks (white and light green parts only) into 1/2-inch half-moons. Mince garlic, strip thyme leaves from stems, and have lemon zest and bay leaves ready. Heat 4 tablespoons olive oil in a large pot over medium heat, then add the carrots, celery, leeks, and 1/2 teaspoon salt. Cook for 5 minutes, stirring occasionally, until the vegetables begin to soften and release their aromatics—this builds a flavorful foundation for the broth.
Add the minced garlic to the vegetables and cook for 30 seconds until fragrant. Pour in 8 cups chicken broth, then add the thyme, lemon zest, bay leaves, and chicken breasts. Bring to a gentle simmer over medium-high heat, then reduce heat to medium-low and simmer for 20-25 minutes until the chicken is cooked through (internal temperature of 165°F). I like to use fresh thyme here because the volatile oils are more pronounced than dried, giving the broth a brighter, more herbaceous flavor.
Remove the chicken breasts from the pot and set aside to cool slightly. Once cool enough to handle, shred the meat with two forks, discarding the skin and bones. Return the pot to a simmer over medium heat and add 1 cup orzo pasta. Cook for 8-10 minutes until the orzo is tender, stirring occasionally to prevent sticking.
Add the shredded chicken back into the pot. Remove from heat and carefully discard the lemon zest strips and bay leaves (a slotted spoon makes this easier). Stir in 1/4 cup fresh lemon juice, 1/4 cup dill, and 1/4 cup fresh parsley. Taste and adjust seasoning as needed. I find that removing the soup from heat before adding the lemon juice preserves the bright, fresh citrus flavor rather than cooking it into the broth.