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1
Preheat an oven to 450 degrees F (230 degrees C). Line a baking sheet with aluminum foil.
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2
Place squash, cut-side up, on the prepared baking sheet. Cover squash with aluminum foil.
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3
Bake in the preheated oven until squash is tender, 45 minutes to 1 hour 15 minutes.
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4
Heat 1 1/2 teaspoons olive oil in a skillet over medium heat; cook and stir mushrooms and garlic in the hot oil until mushrooms are tender, 5 to 10 minutes. Remove skillet from heat.
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5
Heat remaining 1 1/2 teaspoons olive oil in a separate skillet over medium heat; cook and stir onion in the hot oil until softened, about 5 minutes. Add spinach to onion; cook until spinach is heated through, 3 to 5 minutes. Remove skillet from heat.
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6
Heat 6 cups milk in a 5-quart saucepan over medium heat; cook and stir until milk is simmering, about 5 minutes. Whisk remaining 1 cup milk and cornstarch together in a bowl until smooth; stir into simmering milk. Bring milk mixture to just under a boil, stirring constantly with a wooden spoon and scraping bottom of pot to prevent scorching, until milk mixture has thickened into a sauce, about 5 minutes; remove pot from heat.
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7
Whisk 1 1/2 cups Parmesan cheese, honey, nutmeg, sage, salt, and pepper into sauce until smooth. Cover pot with a lid and keep warm over low heat.
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8
Bring a large pot of lightly salted water to a boil. Cook lasagna noodles in the boiling water, stirring occasionally until cooked through but firm to the bite, about 8 minutes. Drain.
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9
Reduce oven temperature to 375 degrees F (190 degrees C).
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10
Scrape flesh from butternut squash into a bowl; stir with a fork until evenly mashed. Stir ginger, salt, and pepper into mashed butternut squash.
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11
Spoon about 1 cup white sauce into the bottom of a 9x13-inch baking dish; arrange a layer of noodles atop white sauce. Layer 1/2 butternut squash mixture, 1/5 remaining white sauce, noodles, spinach mixture, 1/5 white sauce, noodles, 1/2 the mozzarella cheese, mushroom mixture, 1/5 white sauce, noodles, remaining 1/2 butternut squash mixture, 1/5 the white sauce, remaining noodles, remaining 1/5 white sauce, remaining mozzarella cheese, 1/2 cup Parmesan cheese, respectively. Cover dish with aluminum foil and place dish on a baking sheet.
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12
Bake in the preheated oven for 30 minutes. Remove cover and bake until bubbling and cheese is browned, about 15 more minutes. Let stand for 10 minutes before serving.
