Add oil to a large pot over medium heat. When hot, add the onion, celery, carrots, and leek. Cook until the vegetables are softened, about 12 minutes. Watch your heat so the vegetables don't burn or cook too quickly. If needed, turn the heat down. Stir in the garlic, and cook another 30−60 seconds. Add thyme, bay leaf, salt, and pepper.
Remove skin and extra fat from chicken breast. Cut chicken breast crosswise through the bone into two pieces with a chef's knife. Add the broth and the chicken breast to the pot. Bring to almost a boil, turn down to low, add a lid to the pot and simmer until chicken is cooked through, about 15 minutes (to 165°F).
Remove the chicken from the pot. Add the pasta (if using), turn the heat up a little and simmer until pasta is tender. Time depends on your pasta (about 10 minutes). When the chicken is cool enough to handle, shred the meat and add back to the pot when the pasta is done. Remove the bay leaf. Add parsley. Ladle into warm bowls to serve.
