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Special equipment: a broiler-safe 2-quart baking dish
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Position an oven rack at the center of the oven, and preheat the broiler.
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Bring a large pot of water to a boil and cook the pasta according to the package instructions.
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Reserve 3/4 cup of the pasta water; strain the pasta.
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Wipe out the pot.
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Add the butter, and melt over medium heat.
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Add the shallots, and cook, stirring occasionally, until soft, 2 to 4 minutes.
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Add the garlic, and cook, stirring, until soft, about 2 minutes.
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Add the flour, and whisk constantly until it begins to toast, about 2 minutes.
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(The flour will form a thick paste, but that's okay; just keep it moving in the pot.)
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Whisk in the milk and the reserved pasta water, and whisk constantly until the mixture is thick like gravy, 3 to 5 minutes.
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Whisk in the sour cream, 2/3 cup of the mozzarella, the Swiss cheese and 1 tablespoon of the Parmesan.
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Once the cheeses have melted, add the spinach, and cook until just wilted, about 1 minute.
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Stir in the artichoke hearts, pasta, 1 teaspoon salt and 1/4 teaspoon pepper.
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Transfer to a broiler-safe 2-quart baking dish.
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Stir together the remaining 1/3 cup mozzarella and 3 tablespoons Parmesan in a small bowl.
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Sprinkle over the pasta.
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Broil until golden and bubbly, about 2 minutes.