Bring 6 quarts of water to a boil and add 2 tablespoons salt.
In a large mixing bowl, mix the squash, eggs, cheese and nutmeg together until smooth and homogenous. Season with salt and pepper and set aside. Cut the past sheets into 3-inch squares. Place 1 teaspoon squash filling in the center of each, fold to form triangles, pressing firmly around all edges. Take 2 further corners and bring together and press firmly to form little "hats." Place capellaci in boiling water and lower heat to high simmer. Cook until tender, about 2 to 3 minutes.
Melt butter in a 12 to 14-inch saute pan and add sage. Carefully drain the pasta and place in the pan with the butter. Warm gently 1 minute and serve. Sprinkle with the remaining cheese and grate amaretti over using a cheese grater.
Mound 3 1/2 cups of the flour in the center of a large wooden cutting board. Make a well in the middle of the flour and add the eggs and the olive oil. Using a fork, beat together the eggs and oil and begin to incorporate the flour, starting with the inner rim of the well.
As you expand the well, keep pushing the flour up from the base of the mound to retain the well shape. The dough will come together when half of the flour is incorporated.
Start kneading the dough with both hands, using the palms of your hands. Once you have a cohesive mass, remove the dough from the board and scrape up and discard any leftover bits. Lightly reflour the board and continue kneading for six more minutes. The dough should be elastic and a little sticky. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature. Roll or shape as desired.