Adjust an oven rack to the middle position and preheat the oven to 400°F (204°C). Grease a 2-quart casserole dish with unsalted butter and set aside.

In a large mixing bowl, combine the grated Gruyère (or Comté) and Parmigiano-Reggiano. Transfer approximately one-third of this cheese mixture to a separate bowl and reserve it for topping later.

Add the heavy cream, minced garlic, and chopped thyme to the remaining two-thirds of the cheese mixture. Season generously with kosher salt and black pepper. Stir until fully combined.

Add the sliced potatoes to the cream mixture. Using clean hands, gently toss the potatoes until each slice is thoroughly coated, ensuring that any slices stuck together are separated so the cream mixture can reach all surfaces.

Carefully gather a handful of coated potato slices and align them vertically into a neat stack. Set each stack upright into the prepared casserole dish. Continue arranging the potatoes—working around the edges and toward the center—until the dish is tightly packed. If necessary, slice an additional potato and coat it with the remaining cream mixture to fill any gaps.

Pour the excess cream mixture evenly over the arranged potatoes, ensuring that the liquid rises to approximately halfway up the sides of the dish. Not all excess liquid may be required.

Cover the casserole dish tightly with foil and transfer it to the oven. Bake for 30 minutes.

Remove the foil and continue baking for an additional 30 minutes, or until the top of the gratin becomes lightly golden.

Remove the dish from the oven, sprinkle the reserved cheese mixture evenly over the top, and return the dish to the oven. Bake for an additional 30 minutes, or until the surface is deeply golden brown and crisp.

Remove the gratin from the oven and allow it to rest for several minutes before serving. This resting period helps the layers settle and improves sliceability.