Microwave the squash in a covered bowl until tender and easily pierced with a fork, 10 to 15 minutes. Drain the squash well to remove excess moisture and transfer it to a food processor.

Add the butter, sugar, salt, and nutmeg to the food processor and process until the mixture is completely smooth, about 20 seconds.

Transfer the squash to a bowl and stir in the grated Parmesan and pepper. Refrigerate the filling until it is no longer warm, about 20 minutes; chilling the filling makes it firm enough to portion easily.

Dust two large rimmed baking sheets liberally with flour. Divide the pasta dough into five even pieces and keep them covered with plastic wrap so they don't dry out. Roll one piece of dough through a pasta machine into a thin sheet.

Cut the pasta sheet into 4-inch wide rectangles. Place teaspoon-sized dollops of filling along the bottom half of the dough, spaced about 1 1/4 inches apart. If the dough edges feel dry, dab them lightly with water to ensure a good seal.

Fold the top half of the pasta over the filling and press firmly around each mound to remove air pockets and seal the layers. Use a fluted pastry wheel to cut the ravioli apart and trim the edges, then transfer them to the floured baking sheet and cover with a damp towel.

Combine the butter, hazelnuts, sage, and salt in a skillet over medium-high heat. Swirl the pan constantly until the butter is melted, deep golden brown, and smells nutty, about 3 minutes. Remove from the heat and stir in the lemon juice.

Bring a large pot of water to a boil and add a tablespoon of salt. Cook half of the ravioli at a gentle boil until tender, about 2 minutes. Transfer the cooked pasta to a warm platter with a slotted spoon, drizzle with half the sauce, and cover with foil while you cook the second batch.

Swirl two tablespoons of the pasta cooking water into the remaining butter sauce to create a light emulsion, then pour the sauce over all the ravioli. Serve immediately with shaved Parmesan on top.