Preheat the oven to 400 degrees F.
Peel and wash the potatoes and slice them really thinly using a plastic mandoline.
Generously butter a 12-inch ovenproof baking or gratin dish (about 2 1/2 to 3-quarts).
In a small saucepan, bring the half-and-half, garlic cloves and herbes de Provence to a simmer. Season, to taste, with salt and pepper. Remove the garlic cloves. Set aside.
Arrange the potatoes in an overlapping layer on the bottom of the dish. Season the layer generously with salt and pepper. Scatter lightly with some of the grated cheese. Make another potato and cheese layer, finishing with a most generous cheesing.
Press the layers down gently. Pour the half-and-half mixture over the potato layers and fill the dish up to the top of the potatoes. Cover the dish loosely with foil so it is not touching the cheese on top and bake 30 minutes, uncover and bake another 10 to 15 minutes until the cheese is bubbly and nicely browned.
Let cool for 20 to 30 minutes before serving.