Bring a large pot of water to boil and cook chopped turnips until tender, about 15 minutes. Drain well and puree turnips in a food processor or with an immersion blender (or mash by hand).
Heat 2 tablespoons of Olive Oil in a skillet over medium heat and cook shallots, stirring often, until tender and golden brown, about 15 minutes. Remove shallots from skillet and drain them on paper towels. Combine the cream and remaining tablespoon of Olive Oil; bring to a simmer and stir in pureed turnips. Season with nutmeg, salt, and pepper, to taste. Garnish with shallots and parsley before serving.
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