Lightly coat a 4- to 5-quart slow cooker with cooking spray. In the prepared cooker combine soup and sour cream. Stir in 1 1/2 cups shredded cheese, milk, Italian seasoning, and pepper. Stir in potatoes and chicken.
Cover and cook on low-heat setting for 6 to 7 hours. Stir in broccoli. Cover and cook on low-heat setting for 30 minutes more.
Tightly cover slow cooker and tote in an insulated carrier. Tote basil and, if using, additional cheese in an insulated cooler with ice packs. Within 1 hour, plug in slow cooker and keep warm on warm- or low-heat setting. To serve, stir about half of the basil into chicken mixture. Sprinkle with additional cheese ,if desired, and the remaining basil.