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1
Heat the oven to 425 degrees F and arrange a rack in the middle.
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2
Lightly coat a 2-1/2-quart baking dish with butter and set aside.
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3
Heat the olive oil in a small saucepan over medium heat until shimmering, about 3 minutes.
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4
Add the garlic and cook, stirring occasionally, until softened but not browned, about 2 minutes.
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5
Add the cream and salt, season with pepper and a pinch of nutmeg, stir to combine, and cook until the cream is simmering but not boiling; remove from the heat and set aside.
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6
Peel, trim, and slice the parsnips crosswise into 1/8-inch-thick rounds.
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7
Place in a large mixing bowl.
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8
Peel the potatoes, slice crosswise into 1/8-inch-thick rounds, and add to the bowl.
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9
Toss the potatoes and parsnips until evenly combined and set aside.
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10
Evenly layer half of the potato-parsnip mixture in the prepared dish.
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11
Dot with half of the butter and sprinkle with half of the cheese.
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12
Evenly layer the remaining half of the potato-parsnip mixture in the dish and pour the warm cream mixture evenly over the top.
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13
Dot with the remaining butter and sprinkle with the remaining cheese.
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14
Cover the dish with foil and bake until the potatoes and parsnips are tender when pierced with a fork, about 35 to 40 minutes.
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15
Remove the foil and sprinkle the breadcrumbs evenly over the gratin, pressing them down lightly with the back of a spatula (this helps them absorb some of the moisture and keeps them from browning too fast).
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16
Return the gratin to the oven uncovered and bake until the top is golden brown, about 10 minutes more.
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17
Let rest at least 5 minutes before serving.
