Preheat the oven to 400 degrees F (200 degrees C). Place an oven rack in the middle of the oven. Butter a 2-quart casserole dish.

Mix together smoked Cheddar cheese and Parmigiano-Reggiano cheese in a large bowl. Set aside 1/3 of the cheese mixture in a small bowl for topping.

Stir heavy cream, garlic, thyme, kosher salt, and pepper into the large bowl with the cheese mixture. Add sweet potatoes and combine using your hands until every potato slice is fully coated with mixture and not stuck together.

Pick up potato slices a small handful at a time and organize into stacks. Arrange neatly in the prepared casserole dish with potato edges aligned vertically. Begin by placing potato stacks around the perimeter of the dish, working towards the center to create a tightly packed spiral.

Pour leftover cheese and cream mixture over potatoes until mixture comes halfway up the sides of the casserole dish, making sure not to add too much, as it will bubble over. Cover tightly with aluminum foil.

Bake in the preheated oven for 30 minutes.

Remove foil and continue baking until top is golden brown and starting to crisp, about 30 minutes more.

Remove from the oven and sprinkle with remaining smoked Cheddar cheese and Parmigiano-Reggiano cheese. Return to the oven and bake until dark brown and crispy on top, about 30 minutes more.

Remove from the oven. Let cool 5 to 10 minutes, and serve hot.