In a large stockpot or Dutch oven, heat the olive oil over medium heat. Add the chopped onion, celery, and carrots and cook for 5-7 minutes, stirring occasionally, until the vegetables begin to soften.
Add the minced garlic and cook for 1 more minute, stirring constantly, until the garlic becomes fragrant.
Place the whole chicken and cold water into the pot. Add the kosher salt, peppercorns, and bay leaves. Bring the liquid to a boil over high heat. Once boiling, immediately reduce the heat to low to maintain a gentle simmer. Cover the pot with a lid, leaving it slightly ajar. Simmer for 1 hour and 15 minutes, or until the chicken is completely cooked through and tender.
Carefully remove the chicken from the pot and place it on a cutting board. Use two forks to shred all the meat, discarding the skin and bones. Return the shredded chicken meat to the pot.
Increase the heat to medium-high and bring the soup back to a boil. Add the ditalini pasta and cook for 8-10 minutes, stirring occasionally, until the pasta is al dente.
Remove the pot from the heat. Stir in the fresh parsley and lemon juice. Taste and season with additional salt if needed before serving.