Peel the sweet potatoes. Using a mandoline or a sharp knife, slice them about 1/8 inch thick, as evenly as possible.

In a saucepan, melt the butter over medium heat. Add the minced garlic and cook for a minute or two until fragrant. Add in the heavy cream and season with salt and pepper. Heat for a couple of minutes, just until it's steaming with small bubbles around the edge, and don't let it boil hard. (This mixture will not be super thick. But the sauce gets thicker as it cooks due to the potato starch.) Pour it into a measuring cup with a spout.

Shred the Gruyère, Gouda, and Parmesan by hand and toss them together.

Arrange a single layer of sweet potato slices in the baking dish, overlapping them slightly. Season with salt, pepper, and thyme.

Pour a little of the cream mixture over the layer, just enough to lightly coat the slices, then sprinkle with some of the cheese. Repeat the layers. I usually do 3-4 layers. Make sure not to add too much cream to each layer.

Sprinkle the remaining cheese mixture over the top of the dish.

Cover the baking dish tightly with aluminum foil, crimping it around the rim, and bake for about 45 minutes.

Remove the foil and bake for an additional 15-20 minutes, or until a knife slides easily into the center and the top is golden brown and bubbling.

Let it rest for 10 minutes before serving.