Place the potatoes, carrots, celery, onion, garlic, salt, parsley, Italian seasoning, and pepper into a 4-to-6 quart slow cooker. Stir to combine.
In a small bowl, whisk together the broth and cornstarch until smooth. Pour it into the slow cooker along with the cream.
Add the chicken breasts on top, pressing them slightly into the liquid.
Cover and cook on low for 6 to 7 hours, until the chicken is cooked through and the potatoes are fork-tender.
Carefully remove the chicken, shred or chop it, and return it to the slow cooker.
Stir in the frozen peas (if using) and Parmesan cheese. Taste and adjust seasonings as needed.
Ladle into bowls and serve with crusty bread for dipping.