Heat the oil in a large saucepan over medium-high heat. Add the carrots, onion, and celery and cook until softened, 3 to 4 minutes.
Add the chicken broth and 4 cups water to the pan, season with ¾ teaspoon salt and ¼ teaspoon pepper, and bring to a boil. Add the spaghetti, reduce heat, and simmer until tender, 8 to 10 minutes.
Stir in the chicken. Sprinkle with dill and additional pepper.