Preheat over to 375 degrees F. Slice potato, sweet potato, and butternut squash into thin rounds, about 1/8″ slices. Bring a large pot of water to a boil and cook each vegetable separately for 4 minutes, or until almost tender. The vegetables will continue to cook in the oven.
Melt butter in a medium saucepan over medium heat. Add pancetta and cook one minute. Add shallots and garlic and cook another 2 minutes, stirring constantly.
Add flour and stir with whisk. Add thyme. Gradually add milk and 1/2 & 1/2 and stir constantly with a whisk over medium heat until slightly thickened, about 3 minutes. Stir in Parmigiano-Reggiano and blend until melted, about 3 minutes. Season to taste with salt and pepper.
Arrange a layer of potatoes in a buttered 11″ x 7″ casserole dish. Continue to layer casserole with sweet potato slices and butternut squash.
Spoon or pour cream sauce over potato mixture. Top with grated Gruyere cheese and bake at 375 degrees F. for 40 minutes.
Preheat broiler. Broil 3 minutes or until golden brown. Cover with aluminum foil until ready to serve.
Store in airtight container in the refrigerator for up to three days.