Place a Dutch oven on medium-high heat.
Drizzle the 1 tablespoon of Olive oil in the preheated Dutch oven and cook the chicken breast for 8 minutes, before turning them over. Cook for another 8 minutes, or until the internal temperature in the thickest part of the breasts is at a minimum of 165F.
Remove the fully cooked chicken from the Dutch oven and place on a large clean plate.
Turn down the heat to medium and place the butter in the Dutch oven. As it melts, scrape the bottom of the pan to release all of the stuck on pieces.
Add the prepared onion, carrots and garlic into the Dutch oven. Cook until the onions are translucent, 4-5 minutes.
Slowly start pouring in the chicken broth, scraping the bottom of the Dutch oven to release all the flavor. Pour all of it in and bring to a simmer.
Once simmering, turn the heat to low and add the pastina to the broth. Make sure to stir the soup frequently for the first few minutes so that the pasta does not stick to the bottom. Cook for 8 minutes of until the pasta is cooked.
While the pasta is cooking, shred the chicken breasts with the help of 2 forks.
Once the soup is ready, remove it from the heat. Add in the juice of a lemon, and about 1½ cup of shredded chicken. Stir to combine.
Serve the soup hot with optional topping of fresh dill and grated parmesan cheese.