Arrange a rack in the upper third of the oven and heat the oven to 350ºF. Coat a 9x13-inch baking dish with butter.

Prepare the following, adding each to a large pot or Dutch oven: Peel 3 pounds russet potatoes, then cut into thin (⅛-inch) thick slices; finely chop 2 cloves of garlic; add ½ teaspoon fresh thyme (or ¼ teaspoon dried thyme).

Add 3 ½ cups half-and-half and 2 ½ teaspoons kosher salt. Bring to a simmer over medium heat, gently folding the potatoes into the sauce every few minutes, about 13 minutes.

Turn off the heat. Using a slotted spoon, transfer half of the potatoes into the baking dish and arrange into an even layer. Sprinkle with 2 ounces of the shredded cheese. Repeat layering the remaining potatoes, then pour any remaining sauce evenly over the top. Sprinkle with the remaining cheese and scatter 2 tablespoons of butter cut into pieces over the top.

Bake uncovered until the potatoes in the center are tender, and the top is dark golden brown, about 1 hour. Let cool for 10 minutes before serving.