Preheat oven to 350°F (175°C) with the rack in the center position. Generously butter your 9x13-inch baking dish with 1 tablespoon of butter, coating the bottom and sides completely.
Mince the garlic, strip the thyme leaves from their stems, and shred all three cheeses using a box grater. Combine the shredded cheeses in a bowl and set aside.
Set your mandoline to 1/8-inch (3mm) thickness and carefully slice the peeled potatoes into a bowl of cold water to prevent browning. Once all are sliced, drain and pat completely dry with paper towels — removing excess moisture is essential for a creamy, non-watery sauce.
Combine heavy cream, whole milk, minced garlic, thyme, bay leaf, salt, and white pepper in a medium saucepan over medium heat. Bring to a gentle simmer, then reduce heat and steep for 5 minutes; remove from heat and discard the bay leaf.
Arrange one-third of the potato slices in an overlapping pattern in the baking dish, season lightly with salt, pepper, and a tiny dash of nutmeg, then pour one-third of the warm cream sauce over the top and sprinkle with one-third of the cheese. Repeat two more times, ending with cheese on top, then dot with the remaining 1 tablespoon of butter cut into small pieces.
Place the baking dish on a rimmed baking sheet, cover tightly with aluminum foil, and bake for 45 minutes until the potatoes begin to steam and soften.
Carefully remove the foil and continue baking for 30 more minutes, until the top is golden brown and bubbling and the potatoes are completely tender when pierced with a knife.
Remove from the oven and let the dish rest for 15 minutes before serving — this allows the cream sauce to thicken and set so each portion holds together. Serve with a large spoon, making sure to include some of the crispy top with each portion.