Make the sauce (this is a great thing to do in advance):
1 Heat the oil in a large Dutch oven over medium-high heat until shimmering. Add the sage and cook, stirring, until fragrant, about 15 seconds. Add the squash, onion, 2 cups water, 1 teaspoon salt, and several grinds pepper. If the squash pieces aren't submerged, add more water just until they are barely covered.
2 Simmer, uncovered, stirring occasionally, until the squash is very tender, 15 to 20 minutes.
3 Use an immersion blender to blend the mixture directly in the pot until smooth. (Alternatively, transfer to a food processor or blender and blend until smooth). Season generously with salt and pepper to taste. If making ahead, store in the fridge in an airtight container for up to 5 days.
Make the roll-ups:
4 Heat the oven to 425°F. Fill the now-empty pot with water, salt it, and bring to a boil. Add the noodles and cook for 1 minute less than the package directions for al dente. Drain, then lay the noodles in a single layer on a cutting board or baking sheet. (Drizzle with oil if needed to prevent sticking).
6 Meanwhile, heat the oil in a 10-inch oven-safe skillet over medium-high heat until shimmering. Add the mushrooms and cook, stirring occasionally, until deeply browned, 6 to 8 minutes. Add the garlic, sage, red pepper flakes, 1/2 teaspoon of the salt, and several grinds black pepper. Cook until fragrant, about 30 seconds. Add the spinach in handfuls, tearing the leaves into pieces as you go. Cook until wilted, about 1 minute. Remove from the heat.
7 Combine the mozzarella and Parmesan cheeses in a large bowl; set aside 1 cup for topping. Add the cottage cheese, egg, remaining 1/4 teaspoon salt, and several grinds black pepper to the large bowl; stir with a fork to combine. Stir in the sautéed vegetables and season with salt to taste.
8 Divide the cheese mixture evenly over each noodle, spreading until it covers each one. Roll them up, then carefully cut each in half crosswise with kitchen twine or a serrated knife.
9 Spread 3 cups of the squash sauce into the now-empty skillet. Arrange the rolls in the skillet with the ruffled edges facing up. Dollop in any of the filling that may have fallen out when slicing. Sprinkle with the reserved cheese mixture.
10 Bake on the center oven rack until the sauce is bubbling and the cheese is browned, 30 to 35 minutes. Let cool for 5 minutes. Serve leftovers with any extra butternut squash sauce.
