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Peel sweet potatoes or yams and parsnips, and cut into large, consistently sized chunks for even cooking. Place in Dutch oven or other pan with a wide bottom so vegetable chunks are not layered more than two or three deep. Add salt. Add water to cover by 2 inches. Cover, and bring to a boil over high heat. Reduce heat to a gentle boil and cook, covered, for 10 minutes, or until vegetables are about half-cooked. (You can pierce them with a fork, but they’re still quite firm and don’t break apart.) Add apples, replace cover and cook until vegetables are tender (easily pierced by a fork), about 10 more minutes.
Pour vegetable-apple mixture into colander to drain cooking water. Shake gently, then return mixture to hot pan; allow to steam dry for about 5 minutes.
In a separate pan, heat butter and milk until butter is just melted. Pour over vegetable-apple mixture, and mash by hand, using a potato masher or pastry blender, until mixture is fairly smooth but some chunks still remain. Pour into buttered, 1- or 11⁄2-quart casserole. Top with white cheddar cheese, and dust, to taste, with nutmeg and/or black pepper. (Be sparing with the nutmeg, as a little goes a long way.) Place under broiler just until cheese is melted and bubbly.
Makes 6 to 8 servings.