Cook the lasagna noodles. Cook noodles according to package directions until al dente. Drain, then lay flat on a rimmed baking sheet. Lightly spray with cooking spray or drizzle with oil to prevent sticking if they overlap.
Make the filling. In a large bowl, combine 1 cup butternut squash purée, ¾ cup ricotta, ½ cup grated Parm, 1 egg, 2 Tbsp chopped sage, 1 tsp Kosher salt, and ¼ tsp black pepper. Stir until smooth. If the mixture feels too thick to spread, mix in the remaining ½ cup ricotta.
Make the béchamel. In a medium saucepan, melt 4 Tbsp butter over medium heat. Add 2 Tbsp all-purpose flour and whisk until for about 3 minutes, until lightly golden. Slowly whisk in 2-3 cups of warm milk, then increase the heat to medium-high. Bring to a gentle simmer and cook, whisking often, for 3 minutes. Reduce heat to low, then add a few fresh sage leaves and 2 smashed garlic cloves. Cook until thick and velvety, about 8-10 minutes. Stir in ¼ cup grated Parmesan, ⅛ tsp grated nutmeg, a pinch of cayenne, ½ tsp Kosher salt, and freshly cracked black pepper to taste. Set aside.
Assemble the roll-ups. Preheat oven to 375°F (190°C). Lay out the noodles on a clean surface. Spread 2-3 Tbsp filling over each noodle, then sprinkle with 1-2 Tbsp shredded mozzarella. Roll up tightly from one end to the other and set aside. Repeat with remaining noodles and filling.
Bake. Remove and discard the sage leaves and garlic from the béchamel. Spread half of the sauce in the bottom of a 9x13-inch (or similar) baking dish. Arrange the roll-ups in the dish -- either laying flat or standing on their sides. Spoon the remaining béchamel over the top, then sprinkle with remaining mozzarella. Bake for 30 minutes, or until bubbly and golden at the edges.
Garnish, then serve. Top with grated Parmesan, freshly cracked black pepper, and a few crispy sage leaves, if desired. Serve warm.
Optional, crispy fried sage leaves: Heat about 1 inch neutral oil in a small skillet over medium-high heat. When hot, add fresh sage leaves a few at a time and fry for 20–30 seconds, flipping once, until bubbling subsides. Remove with a slotted spoon and drain on paper towels. Sprinkle lightly with kosher salt, then use to garnish the lasagna roll-ups.