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1. Heat the oven to 400° F. Put the milk in a small heavy saucepan and peel and smash one of the garlic cloves. Add it to the milk and then heat the milk over low heat until it just starts to bubble at the edges. Remove from the heat, add the nutmeg and let steep while you continue with the recipe.
2. Peel the second garlic clove, cut it in half and rub the cut side around the inside of a 6-cup baking dish no more than 2 inches deep. Rub 1 tablespoon of the oil all over the inside of the dish.
3. Peel the squash, potatoes and parsnips and cut them into very thin slices (1/8-inch thick). If you have a mandoline, now's the time to use it.
4. Layer the vegetables into the baking dish, alternating between squash, potato and parsnip, and fanning them into concentric, overlapping circles. Season generously with salt and pepper and sprinkle a third of the cheese and a third of the chopped herbs over the slices. Repeat twice, making the top layer as neat and tidy as you can.
5. Remove the garlic clove from the hot milk and pour the milk evenly over the vegetables. Drizzle the remaining olive oil over the top of the gratin and bake for about 50 minutes, until the top is browned and bubbly and the vegetables yield easily when you poke them with a sharp knife. If the vegetables are tender but the top isn't as brown as you'd like, turn on the broiler for a couple of minutes -- watch it carefully so it doesn't burn! Let the gratin cool for at least 5 minutes before serving.
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