Heat the oil and unsalted butter in a saucepan over medium-low heat until it reaches 220 degrees F. Reduce the heat to low, add the shallots, and cook until they are a rich golden brown, 30 to 40 minutes. The temperature should stay below 260 degrees F. Stir the shallots occasionally to make sure they brown evenly. Remove them from the oil with a slotted spoon, drain well, and spread out to cool on paper towels. Once they have dried and crisped, they can be stored at room temperature, covered, for several days.

Peel the turnips to remove the waxy skins and cut them into generous 1-inch chunks. Place them in a saucepan with water to cover and 1 teaspoon of salt. Bring to a boil and simmer, covered, until easily pierced by a paring knife, about 35 minutes. Drain.

In a separate saucepan, heat the milk and salted butter over low heat until the butter has melted and the milk just begins to simmer.

Puree the turnips in several batches in a food processor fitted with the steel blade. With the motor running, add the melted butter and milk in a steady stream. The turnips should be smooth.

Return the puree to the saucepan, season with 1 teaspoon salt and the pepper, and reheat, stirring, over medium heat. Serve piping hot, sprinkled generously with crispy shallots.