Heat oven to 350 degrees F. Rub a baking sheet with olive oil. Place squash cut-side down on baking sheet. Bake until the squash is tender but not brown, about 45 to 60 minutes. Remove from oven and let cool.
Peel squash, discarding skin. Transfer squash to a colander. Top with a plate and a heavy object to weight the squash until it no longer gives off any liquid, about 5 hours the refrigerator.
Measure 2 cups squash. Reserve any remaining squash for another use. Transfer squash to a medium bowl. Using a fork, mash squash to remove lumps. Add cheese, cookies, pepper, and nutmeg. Mix to combine. Reserve 1/4 cup filling.
Lay a sheet of pasta on a work surface. Using a spray bottle, spray squares with a light mist of water. Cut the pasta into 3-inch squares with a knife or pizza wheel. Top each square with 1 teaspoon filling. Fold each square into a triangle to enclose filling. Press edges lightly to seal. With the apex of the triangle pointing away from you, gently pick up the triangle and wrap the 2 points around your index finger. Stretch the triangle just enough so that the 2 ends form a cross. Press the 2 ends together with your thumb to secure. Transfer to a parchment-lined baking sheet. Repeat with remaining pasta. If not cooking immediately, place on a lightly floured baking sheet and refrigerate for up to 1 or 2 days.
Prepare a double boiler. In top of double boiler, combine butter and 1/4 cup water. When butter has melted, add reserved filling and whisk to combine thoroughly to form an emulsion. Keep warm.
Bring a large pot of salted water to a boil. Add cappellacci and cook until the centers are soft and the folded sections are al dente. Remove with a strainer to a large bowl. Add sauce and stir gently with a wooden spoon to combine. If the sauce is watery, rewarm. Stir until thickened.
Place 2 slices of prosciutto on each of 6 plates. Top each with 12 to 16 cappellacci. Sprinkle with nuts, additional cheese, and pepper. Serve immediately.
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