Preheat oven to 375°F. In a small bowl, combine butter and oil.
Brush bottom of a round 9-inch baking dish with some butter mixture. Using a mandoline, slice potatoes very thinly crosswise.
Arrange potato slices vertically in dish. Wedge shallots throughout. Sprinkle with salt and red pepper flakes (if using); brush with remaining butter mixture.
Bake 1 ¼ hours. Add thyme and bake until potatoes are cooked through with a crisp top, about 35 minutes more.
