Preheat oven to 400°F.
Combine cheeses in a large bowl. Transfer 1/3 of cheese mixture to a separate bowl and set aside. Add cream, garlic, shallots, and thyme to cheese mixture. Season generously with salt and pepper. Add potato slices and toss with hands, making sure to get this creamy goodness all over ever single slice of thin potatoes. Yes every single one. It is worth it!
Grease a 2-quart casserole dish with butter. Pick up a handful of potatoes, organizing them into a neat stack, and lay them in the casserole dish with their edges aligned vertically. Continue placing potatoes in casserole, working around the perimeter and into the center until all potatoes have been added. It’s a rustic dish and we are not going for perfection here, we’re going for deliciousness and that will def happen. Potatoes need to be very tightly packed, which means I often sneak more in once the casserole is full, sticking them in between any slices I can. Pour some of the excess cream/cheese mixture evenly over potatoes, until the mixture comes half way up the sides of the casserole. Reserve the excess cream mixture.
Cover tightly with foil and transfer to oven. Bake for 30 minutes. Remove foil, pour as much of the excess cream mixture as ya can without making a huge mess and continue baking until top is pale golden brown, about 30 minutes longer. Carefully remove from oven, pour any more excess cream mixture if there’s any leftover that you couldn’t fit earlier, sprinkle with remaining cheese, and return to oven. Bake until deep golden brown and crisp on top, about 30 minutes longer. Remove from oven, let rest for a few minutes, and serve.