Preheat the oven to 300 F
Take a large (2 ½ quart should work) heavy-bottomed dish or an ovenproof gratin dish (a Le Creuset is ideal), and grease it will with some of the butter, then rub the ½ sliced garlic on the bottom and sides.
Put the cream, milk, and a pinch of salt and pepper, sprigs of thyme and crushed garlic in a sauce pan and bring them gently to a boil. 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. Add a small handful of cheese on each layer. You’ll want to leave about ½ cup of cheese for the final top layer.
Sprinkle the remaining cheese on the top of the last layer.
Pour warm, strained cream over the potatoes. Gently press the potatoes down with your fingers to submerge them in the cream. It’s okay if they’re not 100% covered by the cream.
Dot the top of the gratin with the remaining butter (about ½ tablespoon). Bake for 90 minutes, or until the potatoes are soft, and the top is crispy and golden brown.
Garnish with chopped parsley sprinkled on top (optional)
Serve nice and hot.