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Preheat the oven to 350 degrees F.
The lasagna will resemble ribbon candy when finished. To assemble the lasagna, place the pasta sheet on your work surface. Coat the first 4 inches of the pasta with the squash filling and fold the long end of the pasta over the layer. Spread the next side of the pasta with a thin layer of the filling, fold over the pasta sheet, again about 4 inches, to cover the bottom layers. Sprinkle with pecorino and fold over the pasta again. Repeat until you have used the full length of the pasta sheet. Sprinkle the top with Pecorino and a sage leaf. Transfer to a greased sheet pan. Repeat for the remaining 3 sheets of pasta to make 4 "free form" lasagnas. Place the pan in the oven to bake for 20 minutes.
Cut the ball of dough in half, cover and reserve the piece you are not immediately using to prevent it from drying out. Dust the counter and dough with a little flour. Form the dough into a rectangle and roll it through a pasta machine, 2 or 3 times, at widest setting. Pull and stretch the sheet of dough with the palm of your hand as it emerges from the rollers. Reduce the setting and crank the dough through again, 2 or 3 times. Continue tightening until the machine is at the narrowest setting; the dough should be paper-thin, about 1/8-inch thick (keep in mind, overly thick pasta tastes gummy.) Make 4 pieces, about 20-inches long and 6-inches wide.