Cut any potatoes that are larger than a ping pong ball into quarters; if smaller than a ping pong ball, cut in half. You want the pieces to be a rough 1-inch dice. Lay them in a 6-quart or larger slow cooker.
Stir the carrots into the slow cooker with the potatoes, then spread them into an even layer. Arrange the chicken in the center, placing it on top of the vegetables.
In a small saucepan, melt the butter over medium-low heat. Remove from the heat and stir in the garlic, lemon juice and zest, Italian seasoning, salt, and pepper. Pour the mixture over the top of the chicken and vegetables.
Cover the slow cooker. Cook on HIGH for 1 1/2 to 2 hours or low for 4 to 5 hours. (The total amount of time you need will vary based upon your slow cooker model and the size of your chicken breasts. If your slow cooker runs hot—mine often does—check early to avoid drying out the meat). The chicken is done when it reaches 165 degrees F at the center with an instant read thermometer. The moment the chicken is done cooking, remove it to a plate, and cover to keep warm. Test the vegetables to see if they are tender. If they are not yet tender, give them a stir, recover the slow cooker, then continue cooking until they are tender and pierce easily with a fork, up to 1 additional hour on high or 2 hours on low. When ready to serve, return the chicken to the slow cooker. Sprinkle the Parmesan and parsley over the top. Enjoy warm.
TO STORE: Place cooked leftovers in an airtight storage container in the refrigerator for up to 4 days.
TO REHEAT: Gently rewarm chicken and potatoes on a plate in the microwave until warmed through.
TO FREEZE: While I do not recommend freezing the potatoes (they can become mealy when frozen), you can freeze the chicken and carrots. Store cooked chicken and carrots in an airtight freezer-safe container in the freezer for up to 3 months. Let thaw in the refrigerator overnight before reheating.