Preheat oven to 400°F with rack in middle position. Toss together squash, olive oil, and cayenne pepper on a large rimmed baking sheet; spread into an even layer. Bake until tender, about 35 minutes.

Meanwhile, cook butter in a small saucepan over medium until butter browns and has a nutty aroma, 4 to 5 minutes. Remove from heat.

Process ricotta, Parmesan, eggs, and salt in a food processor until well combined, about 45 seconds. Transfer cheese mixture to a medium bowl, and stir in spinach and chopped sage; set aside. Clean food processor bowl.

Add cooked squash to the clean food processor bowl. Process until smooth, about 1 minute, stopping to scrape sides halfway through. Add brown butter to squash, and pulse until combined, about 10 pulses. Transfer to a bowl.

Lightly grease a 13- x 9-inch baking dish with butter. Spread ¾ cup of the squash puree in bottom of prepared dish. Arrange 3 noodles on top. Spread 1 ¾ cups of the ricotta mixture over noodles, top with ¾ cup of the squash puree, and sprinkle with 1 ⅓ cups of the mozzarella. Repeat the layering process twice, beginning with noodles and ending with mozzarella. Cover lasagna with lightly greased aluminum foil. (If desired, wrap entire freezer-safe baking dish in plastic wrap and then foil, and freeze up to 3 months. Thaw in refrigerator 24 hours. Remove the plastic wrap, and cover with lightly greased foil before baking.)

Place a baking sheet lined with foil on bottom of oven to catch any cheese drips. Bake lasagna at 400°F for 30 minutes (45 minutes if baking from thawed). Uncover; bake until the lasagna is heated through and top is nicely browned, 20 to 25 more minutes. Let stand 10 minutes before serving. Garnish with additional fresh sage leaves.