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Cook parsnips in a 5-to 6-quart pot of boiling water with 1 tablespoon salt, covered, until very tender, 15 to 20 minutes, then drain.
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Purée hot parsnips with butter, milk, and 1/2 teaspoon each of salt and pepper in a food processor until smooth. Season with salt, then transfer to a serving dish and keep warm, covered.
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Meanwhile, remove and reserve all but smallest leaves from Brussels sprouts, trimming stem ends as necessary.
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Heat oil in a 12-inch heavy skillet over medium-high heat until it shimmers, then cook leaves, tossing occasionally, until browned in patches, 2 to 3 minutes. Add water and cook, tossing, until leaves are slightly wilted and water has evaporated, 1 to 2 minutes. Season with salt and scatter leaves over parsnip purée.