Heat oven to 350 F with the rack in the middle position, and grease a 2 1/2 quart soufflé dish with the butter.
Place the cream and half and half in a 4 quart saucepan. Tie 5 of the thyme sprigs and 2 of the rosemary sprigs with butcher's twine and toss it in the cream mixture along with the garlic. Turn the heat to medium and bring to a bare simmer (180 F), then kill the heat. Add 1/2 teaspoon of the salt, 1/2 teaspoon of the white pepper, and the nutmeg. Cover tightly with a lid and let steep for 1 hour. Uncover the pot, remove the herbs and set the steeped mixture aside.
Combine the Parmesan and Asiago in a small bowl. Strip and finely chop the remaining thyme and rosemary and add to the cheese. Divide the mixture in half and place in two separate bowls. Half of the mixture will go into the layers and the other half will be for topping the gratin.
Slice the potatoes 1/8 inch thick using a mandoline or the slicing attachment of a food processor.
Make the first layer of the potatoes by slightly overlapping each slice to cover the bottom of the dish. Sprinkle 1/4 teaspoon of the salt, 1/4 teaspoon of the pepper, and 2 heaping tablespoons of the cheese herb mixture over the whole layer. Continue this process of layering, seasoning, and cheesing until you run out of everything except the reserved half of the cheese. You should end up with 4 layers.
Pour the cream mixture over and press on the potatoes to submerge with a spatula or wooden spoon. Top with the remaining cheese and cover the dish with foil. Bake for 1 hour, then remove the foil, crank the heat to 425 F, and continue cooking until the top is golden brown and bubbling, 25 minutes.
Remove the gratin to a cooling rack and let it rest for a minimum of 30 minutes before serving.