Preheat the oven to 350°F.
Take a large, heavy-bottomed dish or an ovenproof gratin dish, and grease it well with the butter, then rub the ½ sliced garlic on the bottom and sides. Discard the garlic.
Put the cream, milk, and a pinch of salt and pepper and sprigs of thyme in a saucepan. Using a garlic press, press 1 clove of garlic into the sauce (or mince finely, and add to the sauce).
Bring gently to a boil. Lower heat and simmer for a couple of minutes. Remove from heat, then strain into a bowl, and discard the thyme and garlic. Set aside briefly.
Layer the potato slices in the dish, slightly overlapping each other, seasoning each layer with the salt and pepper. Once one layer is added, ladle enough of the strained cream sauce to coat the potatoes. Repeat with the process until out of potatoes.
Add a little more cream, if necessary. The cream should just barely be peaking around the edges of the potatoes when you press down on them (resist the temptation to add too much cream!). You may have extra potato slices, or you may have a little extra cream.
Sprinkle the cheese on the top of the last layer.
Bake for 45 minutes and then check them, you'll most likely need to cover them with a piece of foil to keep them from browning anymore. Bake for another 15 to 20 minutes, or until a sharp knife easily pierces the potatoes with absolutely no resistance.
Remove from oven and let rest for 5 to 10 minutes. Garnish with chopped parsley sprinkled on top.