Preheat oven to 375 degrees. In a small bowl, combine butter and oil. Brush bottom of a round 9-inch baking dish with some butter mixture. With a sharp knife or 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 1/4 hours. Add thyme and bake until potatoes are cooked through with a crisp top, about 35 minutes more.
The thinner the potatoes are cut, the crispier they'll become in the oven. Keep the slices together as you cut them so that you can arrange them easily in the dish.