Melt the butter over medium heat in a shallow pot or large saute pan. Add the fennel, celery root, potatoes, apples, 1 1/2 teaspoons salt and 1/2 teaspoon pepper. Saute the vegetables, stirring occasionally, until they begin to soften, about 4 to 5 minutes. Add the cider and tightly cover the pot. Simmer over low heat (I pull the pot halfway off the heat) for 30 to 40 minutes, stirring occasionally, until the vegetables are very soft. If the vegetables begin to burn or they seem dry, add another few tablespoons of apple cider or some water.
When the vegetables are cooked, add the cream and cook for 1 more minute. Transfer the mixture to a food mill fitted with the coarsest blade and process. (You can also use a food processor but the texture will be smoother than with the food mill.) Taste for salt and pepper and return to the pot to keep warm. Serve warm.
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