Transfer ⅓ of the cheese mixture to a separate bowl, and set it aside.
Add the cream, garlic, and thyme to the cheese mixture. Season it generously with salt and pepper.
Add the potato slices, and manually toss them (making sure to separate any slices sticking together) until all slices are thoroughly coated with the cream mixture.
Grease a 2-quart casserole dish with butter.
Pick up a handful of potato slices, organizing them into a neat stack, and lay them in the casserole dish with their edges aligned vertically like a roll of coins.
Continue placing potatoes in the dish, working around the perimeter and into the center until all the potatoes have been added.
The potatoes should be very tightly packed. If necessary, slice an additional potato, coat the slices with the cream mixture, and add them to the casserole. Pour the excess cream/cheese mixture evenly over the potatoes until it comes halfway up the sides of the casserole. (You may not need to use all the excess liquid.)
Cover the dish tightly with foil, and transfer it to the oven. Bake it for 30 minutes.
Remove the foil, and continue baking until the top is pale golden brown, about 30 minutes longer.
Carefully remove the dish from the oven, and sprinkle it with the remaining cheese. Return it to the oven for an additional 30 minutes or so, until it’s deep golden brown and crisp on top.
Remove your Cheesy Hasselback Potatoes Gratin from the oven, and let them rest for a few minutes. Serve, and enjoy!