Preheat oven to 400 degrees F (200 degrees C). Grease a baking dish.
Beat cream cheese, hot sauce, 1/2 cup pepper Jack cheese, and blue cheese dressing together in a large bowl; fold in chicken until completely coated. Pour mixture into the prepared baking dish and top with crumbled blue cheese and 1/3 cup pepper Jack cheese.
Bake in the preheated oven until cheese is melted and dip is bubbling up the sides, 15 to 20 minutes.