Place half-and-half, garlic, and thyme in a saucepan; bring to a simmer and keep warm. Before using, remove garlic clove and thyme stem. Peel the potatoes and thinly slice. Oil a ceramic baking dish. Arrange potato slices on bottom of dish, overlapping edges. Season potatoes with salt and pepper; cover with one-third of the heated half-and-half and top with one third of the cheese and a sprinkle of nutmeg. Repeat with two more layers. Top gratin with reserved cheese. Bake in a wood-burning oven for 40–45 minutes, or until potatoes are tender when pierced with the tip of a knife and the top is golden brown. If top is over-browning, cover very loosely with foil. Remove and let rest for 15 minutes before cutting to serve.