Add the squash, shallot and garlic to a baking sheet. Drizzle with the olive oil and toss to combine. Sprinkle with the thyme, sage, 1/4 teaspoon salt and a few grinds of black pepper and toss again. Roast until the squash is tender and light golden, about 30 minutes. Let cool.
Meanwhile, bring a large pot of water to a boil; add salt. Add the pasta shells and cook until al dente according to the package directions. Drain and let cool.
Squeeze the roasted garlic into a small food processor or blender, add the squash, shallot and vegetable broth and puree until smooth.
Mix together the spinach, cream cheese, ricotta, red pepper flakes, 2/3 cup of the Parmesan and 1 tablespoon salt in a large bowl. Stuff about 2 tablespoons of the spinach mixture into each shell; set aside.
Add the pureed squash mixture, heavy cream and butter to a large skillet over medium heat. Cook, whisking frequently, until thickened, about 5 minutes. Remove from the heat and whisk in the nutmeg, 1 1/2 teaspoons salt and the remaining 1 1/3 cups Parmesan.
Spread the cheese sauce into the bottom of a 9-by-13-inch baking dish. Arrange the stuffed shells in the baking dish seam-side up, shingling them so they all fit (you should have 5 shells along the short side and 6 shells along the long side). Top with the mozzarella. Bake until the cheese is melted and bubbling, 15 to 20 minutes. Top with more Parmesan and sage before serving.
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