Take the chicken out of the refrigerator about 20 minutes before cooking so it can come to room temperature. Preheat the oven to 425 F. Generously season the chicken on both sides with salt and pepper.

In a large, ovenproof skillet, heat 1 tablespoon of the olive oil over high heat. When the oil slides easily across the pan, add the chicken skin-side down and cook, untouched, for 1 minute. Transfer the skillet to the oven and cook until the chicken shows no sign of pink when pierced near the bone, about 30 minutes. Flip each breast and set the skillet aside until the chicken is cool enough to handle. Chop or shred the meat into bite-size pieces (these can go back in the skillet), discarding the skin and bones.

In a large pot, heat the remaining 2 tablespoons of olive oil over high heat. When the oil shimmers, add the carrots, celery, onion, and a pinch of salt, and cook for 1 minute. Reduce the heat to medium, cover, and cook for 15 minutes, stirring occasionally.

Pour in the chicken broth and water and bring to a boil. Reduce the heat and simmer until the vegetables are tender, about 15 minutes. Add the chicken (including the juices it released in the skillet), cooked pasta, dill and simmer for 5 minutes more. Add salt to taste.

Serve and drizzle with additional olive oil as desired.