Preheat your oven to 375 degrees. Butter a 2.5 to 3 quart baking dish and set it aside. Peel the potatoes and use a mandoline set to about 1/8 inch, or a very sharp knife, to slice them into even rounds. Try to keep the thickness consistent so the layers cook at the same rate. Place the sliced potatoes in a bowl of cold water to keep them from browning while you make the sauce.

Melt the butter in a medium saucepan over medium heat. Add the minced garlic and cook for about 1 minute, stirring constantly, until it smells fragrant but has not browned. Whisk in the flour and keep whisking for about 1 minute to cook out the raw flour taste. You should have a pale, thick paste in the pan.

Slowly pour in the heavy cream and milk, whisking constantly as you add it so no lumps form. Increase the heat slightly and continue whisking until the sauce comes to a gentle simmer and thickens enough to coat the back of a spoon, about 4 to 5 minutes. Stir in the thyme leaves, salt, and black pepper. Remove the pan from the heat. Stir in 1 cup of the shredded cheddar until melted and smooth.

Drain the potatoes and pat them dry with a clean kitchen towel. This step matters because extra moisture on the potatoes will thin out your sauce. Arrange about one third of the potato slices in an even layer on the bottom of the prepared baking dish, overlapping them slightly like shingles. Spoon about one third of the cream sauce over the potatoes and spread it to the edges. Sprinkle with about 1/2 cup of the remaining shredded cheddar. Repeat the layers two more times, finishing with the remaining cheddar on top.

Cover the dish tightly with aluminum foil, making sure the foil does not touch the cheese on top. Bake covered at 375 degrees for 45 minutes. At this point the potatoes should be just beginning to become tender and the sauce should be bubbling around the edges.

Remove the foil and continue baking for another 20 to 25 minutes until the top is deep golden brown and the potatoes are completely tender when a sharp knife or fork slides through the center with no resistance. If the top is browning faster than the potatoes are cooking, lay the foil loosely back on top for the last 10 minutes.

Pull the dish from the oven and let it rest for 10 minutes before serving. This resting time lets the sauce firm up slightly so the portions hold together when scooped. Garnish with a few fresh thyme sprigs if you like.