Place the chicken breasts in a large saucepan or pot, large enough for them to fit side by side without overlapping. We use a 4-quart enameled Dutch oven. Cover the chicken with chicken broth, about 2–2¼ cups, and add a pinch of salt.
Start the chicken over high heat and bring the broth to a boil. As soon as it starts boiling, lower the heat to medium-low and cover. Simmer gently for 20–25 minutes, turning halfway through. Make sure the internal temperature of the thickest part of the chicken breast is at least 165°F.
Remove the chicken breasts from the poaching liquid and place them on a cutting board to cool. Strain the cooking liquid through a fine-mesh sieve into a bowl or large measuring cup and reserve.
When the chicken has cooled enough to handle, shred, slice, or cut it into bite-sized cubes. Tip: You can start preparing the soup while the chicken cools, then shred it during the first simmer to save time.
If there’s no liquid left in the pot, there’s no need to wash or wipe it. If it’s wet, pat it dry.
Return the pot to medium heat and add the olive oil. When it’s hot, add the onion and celery. Cook for 3 minutes, stirring often.
Add the garlic, tomato paste, Italian seasoning, and black pepper. Cook for 30–60 seconds, stirring frequently.
Add the reserved chicken broth and the remaining chicken broth while stirring and gently scraping any bits off the bottom of the pot.
Raise the heat and bring the broth to a boil. Lower the heat to medium-low, cover, and simmer gently for 5–7 minutes to let the flavors come together.
Stir in the pasta. If needed, raise the heat to bring the soup back to a simmer. Cover and cook for 5 minutes, until the pasta is almost done. Stir often at first to keep the pasta from sticking to the bottom.
Add the reserved cooked chicken and the mixed vegetables and stir well. Raise the heat to bring the soup back to a simmer. Lower the heat to medium-low, cover, and cook for 5–7 minutes to finish cooking the pasta, heat the soup through, and let the flavors develop further. Stir occasionally.
Taste and add salt if needed. As a reference, we did not add extra to ours. Garnish with chopped parsley, if desired, and serve.