Heat oven to 425°F. Lightly butter a 10-inch round or oval, 2-inch-deep gratin dish or pie plate and set aside. Peel and trim potatoes and parsnips. Slice parsnips crosswise into 1/8-inch-thick rounds, pat dry, and place in a large mixing bowl. Slice potatoes crosswise into 1/8-inch-thick rounds. Toss vegetables together and set aside.
Warm cream in a small saucepan over medium heat until simmering but not boiling; remove from heat. Whisk in garlic purée (oil and all) and salt, and continue whisking until well blended. Season with nutmeg and pepper to taste, then return the pan to low heat.
Place 1/2 of the vegetable mixture in the bottom of the prepared dish. Dot with 2 tablespoons of the butter and sprinkle with 1/4 cup of the cheese. Top with remaining vegetable mixture, dot with remaining butter, and sprinkle with remaining cheese. Pour warm cream over top.
Cover the dish with foil and bake until vegetables are tender when pierced with a fork and the top is bubbling, about 35 minutes.
Remove foil and sprinkle breadcrumbs evenly over gratin, pressing them down lightly with the back of a spatula so they absorb some of the moisture and don’t brown too fast. Return gratin to oven and bake until top is golden brown, about 10 minutes more. Remove gratin from oven and allow to rest at least 5 minutes before serving.
Note: To make roasted garlic purée, peel and smash 6 garlic cloves and place them on a square of aluminum foil. Add a splash of olive oil, fold the foil to make a packet, and roast at 350°F until soft, 15 to 20 minutes.
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