For the pasta, combine the flour, eggs, and olive oil in the bowl of a mixer fitted with the dough hook attachment. Mix until well blended and a smooth, satiny dough forms. Alternatively, you can mix the dough by hand: mound the flour on a flat surface and form a well in the center. Add the eggs, olive oil, and salt to the well. Use a fork to beat the eggs, then gradually work in the flour to mix with the eggs. When the dough begins to form, knead it for about 5 minutes to make a very smooth dough.

Wrap the pasta dough in plastic wrap and refrigerate while you make the filling.

Preheat the oven to 350°F. Drizzle 1 tablespoon of the olive oil over the halved butternut squash, set it in a baking dish cut-side down, and roast until fully tender, about 1 hour.

While the squash is roasting, heat the remaining tablespoon of olive oil in a small skillet over medium heat. Add the onion and sauté until tender, about 5 minutes. Add the garlic and cook for 2 minutes longer, stirring often. Put the onion and garlic in a medium bowl and set aside.

When the squash is tender, set it aside to cool. When cool enough to handle, scrape out the flesh. Add about 2 cups to the bowl with the onion; save the rest for another use. Stir the squash to evenly mix it with the onion, then fold in the goat cheese. Season to taste with salt.

Use a pasta machine to roll out the dough into sheets about 18-inches long.

Lay one of the pasta sheets on the counter and spoon the filling in mounds of about 1 tablespoon onto the dough in 2 rows, spaced 2 inches apart. Lightly brush the beaten egg around each mound of filling and lay the other sheet of pasta on top. Use your fingers to gently press the two sheets of pasta together, pressing out air around the filling to avoid air pockets. Use a pasta cutter or knife to cut between the ravioli into squares. Set aside on a lightly floured surface until ready to cook (the ravioli can be made ahead and frozen for up to a week).

Bring a large pot of salted water to a boil. While the water is heating, melt the butter in a medium skillet over medium heat until it is just lightly browned. Add the apple and sage and cook, stirring, for 1 minute. Set the skillet aside.

When the water’s boiling, add the ravioli and cook for about 5 minutes. Drain well and put the ravioli in individual shallow bowls. Spoon the apple mixture over, top with crab, and sprinkle Parmesan cheese over. Serve right away.