Beat 3 packages cream cheese, Cheddar cheese, and red pepper at medium speed with an electric mixer until blended and smooth. Stir in green onions.
Spread half of cream cheese mixture into an 8-inch round cake pan lined with plastic wrap; spread top with tomato chutney, leaving a 1/2-inch border of cream cheese around outside edge. Spread remaining cream cheese mixture over chutney and cream cheese border. Cover and chill at least 8 hours or up to 24 hours.
Invert cheesecake onto a serving platter; remove plastic wrap. Stir together 1 Tbsp. milk and remaining 1 package cream cheese, stirring until spreading consistency and adding additional milk, if needed. Frost top and sides of cheesecake. Serve with vegetables and crackers. Garnish, if desired.