Prep the Noodles: Bring a large pot of salted water to a boil. Add the lasagna noodles and cook according to package directions until al dente. You want them pliable but not mushy. Drain the noodles and lay them out flat on a clean surface (like parchment paper or clean tea towels) to prevent them from sticking together.
Make the Filling: In a medium bowl, combine the mashed butternut squash (or pumpkin purée), 1 cup of the ricotta, the ½ cup grated Parmesan, the large egg, 2 tablespoons of the finely chopped sage, the 1 teaspoon Kosher salt, and the ¼ teaspoon black pepper. Mix well until everything is thoroughly combined. Stir in ½ cup of the shredded mozzarella cheese.
Whip up the Bechamel: In a medium saucepan, melt the unsalted butter over medium heat. Whisk in the flour and cook for about 1-2 minutes, whisking constantly, until the mixture is bubbly and smells slightly nutty (this is your roux!).
Add the Milk: Gradually whisk in the warm milk, a little at a time, ensuring each addition is incorporated before adding more. Keep whisking continuously to prevent lumps.
Simmer the Sauce: Bring the sauce to a gentle simmer, continuing to whisk occasionally. Add the ¼ cup grated Parmesan, nutmeg, cayenne pepper, ½ teaspoon salt, black pepper to taste, the 4 large sage leaves, and the 2 smashed garlic cloves. Reduce the heat to low and let it simmer gently for 5-10 minutes, allowing the flavors to infuse and the sauce to thicken. Remove the sage leaves and garlic cloves before using.
Assemble the Roll Ups: Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish.
Spread the Filling: Lay one cooked lasagna noodle flat. Spread a generous amount (about 2-3 tablespoons) of the butternut squash filling evenly over the noodle, leaving a small border on the short ends.
Add Cheese: Sprinkle a little of the remaining shredded mozzarella (you set aside from the 1½ cups total) over the filling on the noodle.
Roll 'Em Up: Starting from one short end, carefully and gently roll the noodle up into a spiral.
Arrange and Sauce: Pour about ½ cup of the prepared bechamel sauce into the bottom of the prepared baking dish. Arrange the rolled-up lasagna noodles seam-side down in the dish, snuggled next to each other. Pour the remaining bechamel sauce evenly over the top of the roll ups, making sure they are mostly covered.
Top with Cheese: Sprinkle the remaining shredded mozzarella cheese and the leftover ¼ cup ricotta (in dollops or crumbled) over the top of the sauce and roll ups. You can also add a little extra grated Parmesan here if you like!
Bake: Cover the baking dish tightly with foil. Bake for 25 minutes.
Brown the Top: Remove the foil and bake for another 10-15 minutes, or until the sauce is bubbly and the cheese is melted and golden brown.
(Optional) Make Crispy Sage: While the lasagna is baking, heat about ¼ inch of neutral oil in a small skillet over medium heat until shimmering. Carefully add the fresh sage leaves (they should sizzle). Fry for just 15-30 seconds until crisp. Remove with a slotted spoon and drain on a paper towel-lined plate. Be careful, they fry very quickly!
Rest and Serve: Let the lasagna roll ups rest for 5-10 minutes before serving. This helps them hold their shape. Serve warm, garnished with extra grated Parmesan, a sprinkle of fresh chopped sage, and the crispy sage leaves if you made them. Enjoy that cozy hug!