Microwave the squash in a covered bowl until tender and easily pierced with a fork, 10 to 15 minutes. Drain the squash thoroughly to prevent a soggy filling, then transfer to a food processor.

Add the melted butter, sugar, salt, and nutmeg to the processor and pulse until completely smooth. Transfer the mixture to a bowl, stir in the grated Parmesan and pepper, and refrigerate until cold, about 20 minutes.

Dust two large baking sheets with flour. Divide the pasta dough into five pieces and roll each through a pasta machine into thin sheets, then cut into 4-inch wide rectangles.

Place generous teaspoon-sized dollops of squash filling along the bottom half of the dough rectangles, spacing them 1 1/4 inches apart. Fold the top half of the dough over the filling, dabbing the edges with water if they feel dry, and press firmly around each mound to create a tight seal without air bubbles.

Cut the ravioli apart using a fluted pastry wheel and transfer them to the prepared baking sheets. Keep them covered with a damp towel while you finish the rest of the batch.

Cook the butter, hazelnuts, sage, and salt in a large skillet over medium-high heat. Swirl the pan constantly until the butter melts, turns a deep golden brown, and smells intensely 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 the ravioli in two separate batches to avoid crowding, boiling gently until tender, about 2 minutes.

Use a slotted spoon to transfer the cooked ravioli to a warm platter. Drizzle with some of the brown butter sauce and cover with foil while you cook the second batch. Swirl two tablespoons of the pasta cooking water into the remaining sauce to emulsify it, then pour over the ravioli. Serve immediately with shaved Parmesan.