Arrange a rack in the upper third of the oven and second rack below it. Line the lower rack with aluminum foil to catch any drips. Heat the oven to 450 F. Coat an 18×13-inch rimmed baking sheet (half sheet pan) with unsalted butter. Coat one side of a sheet or two of aluminum foil large enough to cover the baking sheet with butter.

Bring the half-and-half, garlic cloves, kosher salt, and thyme to a simmer over medium heat in a small saucepan, stirring until the salt is dissolved. Turn off the heat.

Peel the russet potatoes, then cut with a chef’s knife or on a mandolin into thin (1/8-inch-thick) slices. Arrange half of the potatoes on the baking sheet in an even layer (some overlapping is okay). Sprinkle evenly with 4 ounces of the shredded Gruyère cheese.

Pour about half of the half-and-half mixture (1 2/3 cups) evenly over the potatoes. Arrange the remaining potatoes evenly over the cheese. Pour the remaining half-and-half mixture evenly over the potatoes (the baking sheet will be very full). Cover the baking sheet tightly with the foil buttered-side down.

Bake for 20 minutes. Uncover and sprinkle with the remaining 4 ounces Gruyère cheese and Parmesan cheese. Return to the oven and bake uncovered until the potatoes in the center are very tender, and the top is well-browned, about 20 minutes more. Sprinkle with fresh chives if desired. Let cool for 5 minutes before serving.