Preheat: Preheat the oven to 375 degrees Fahrenheit.
Boil the squash: Peel and cut the butternut squash into small cubes about 1/2-inch thick. (Go to How to Cut Butternut Squash for a video! A serrated vegetable peeler works best.) Bring a medium pot of water to a boil. Add the squash and cook for about 5 to 6 minutes until fork tender (test a piece to see if it is done). Drain, then place in a medium bowl. Mix with the olive oil, kosher salt and garlic powder.
Meanwhile, boil the noodles*: Bring a large pot of well salted water to a boil. Boil the noodles until just before al dente, about 5 minutes, stirring often. Drain the noodles, then drizzle a baking sheet with a bit of olive oil. Lay the noodles flat onto the sheet, then turn them over so they become coated with olive oil to prevent sticking.
Prepare the ricotta sauce: Add 1 tablespoon chopped sage to a medium bowl and reserve about ½ tablespoon for topping the lasagna. To the bowl, add the lemon zest, nutmeg, ricotta cheese, salt and pepper and milk. Stir.
Layer the lasagna: In a 9” x 13” baking dish, spread ½ cup ricotta sauce on the bottom of the pan. Then top with 1 layer of noodles, ½ of the cooked squash, scant 2/3 cup cheese sauce, 1 cup mozzarella and ⅓ cup Pecorino cheese. Repeat again: 1 layer of noodles (you may need to cut a noodle in half for the side, depending on your pan), the remaining squash, scant 2/3 cup cheese sauce, 1 cup mozzarella cheese and ⅓ cup Pecorino cheese. Finally, top with noodles (again, you may need to cut a noodle in half for the top layer), then the remaining ricotta sauce spread with a spatula. Sprinkle the entire top with the remaining ½ cup mozzarella cheese, ⅓ cup Pecorino cheese, and ½ tablespoon sage.
Bake the lasagna: Cover the pan with aluminum foil and bake for 40 minutes. Carefully remove the foil and bake another 20 minutes, until the top is browned. Let stand for 5 to 10 minutes before serving. Leftovers can be refrigerated for 2 to 3 days and reheated in a 400°F oven. (See make ahead and storage notes below.)