Add olive oil to a large Dutch oven or soup pot over medium heat.
Season chicken pieces with garlic powder, salt, and pepper. Cook chicken on each side for 3 to 4 minutes, or until it's almost cooked through. Set aside to be cut or shredded.
Add chopped onions, carrots, celery, and oregano (and garlic if using) to the pan. Cook over medium heat until vegetables are softened, about 10 to 15 minutes. If vegetables are drying out, add a few splashes of chicken broth to keep the mixture from burning.
When vegetables are tender, add remaining chicken broth and water to the pot. Add the bay leaf, if using. Bring to a boil over high heat, then add the pastina. Stir, then lower the heat a bit (so it doesn't boil over) and cook until pasta is al dente, about 10 minutes.
While the pasta is cooking, shred the chicken.
When the pastina is cooked, add the shredded chicken to the pot along with one cup of grated cheese. Stir to combine. Remove bay leaf if one was used.
Serve with extra parmesan on the side.
Notes
To store your soup, let it cool down first. Separate the pastina from the rest of the soup to keep the pasta from absorbing the liquid and turning mushy. Transfer to airtight containers. You can store the soup in the refrigerator for up to 4 days or in the freezer for up to 3 months. If you plan to freeze the soup, be sure to leave some extra space at the top of the container to allow for expansion as the soup freezes.