Assemble pizzas: Place about 1-2 tbsp of alfredo on each Naan flatbread. Place a single layer of potato over alfredo. Sprinkle ¼ cup of each cheese over potatoes. Layer ¼ of the chanterelles on top of cheese. Sprinkle thyme leaves over chanterelles. Bake on lowest rack in oven to get a nice crispy crust until cheese is melted.
In a medium bowl, quickly toss together arugula and dressing. Remove pizza from oven and top with about ½ cup of the salad. Enjoy!