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1
Slice the shallots crosswise into very thin rings.
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2
In a saucepan, heat the oil with 3 tablespoons butter over medium-low heat until it begins to bubble.
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3
Reduce the heat to low, add the shallots and cook until they are a rich golden brown, about 30-40 minutes; stir the shallots occasionally while they are cooking to make sure they brown evenly.
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4
Remove the shallots from the oil with a slotted spoon and drain on paper towels.
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5
Once the shallots have dried and crisped, in about 15 minutes, they can be stored in a cool place, covered, for several days.
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6
Serve the shallots at room temperature.
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7
Peel the turnips to remove their waxy skins, and cut them into generous 1-inch chunks.
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8
Place them in a saucepan with water to cover and 1 teaspoon of salt.
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9
Bring to a boil and simmer, covered, until easily pierced by a paring knife, about 35 minutes.
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10
In a separate saucepan, heat the milk and 6 tablespoons butter over low heat until the butter has melted and the milk just begins to simmer.
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11
Drain the turnips, and puree (in several batches, if necessary) in a food processor.
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12
With the motor running, add the melted butter and milk in a steady stream; the turnips should be very smooth.
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13
Return the turnip puree to the saucepan, season with 1 teaspoon salt and the pepper and reheat, stirring over a medium flame.
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14
Serve piping hot, sprinkled generously with crispy shallots.
