Place the pancetta in a large skillet over a medium flame. Cook for 5 minutes, until slightly crispy. Toss in the onion and olive oil and cook until translucent, about 3 minutes. Add the garlic, sage, and chestnuts to the pan and cook for 10 minutes, stirring often. Toss in the squash pieces, chicken stock, and freshly grated nutmeg. Cover to steam. Let the squash cook for 20 minutes or until soft. Allow the stock to evaporate and then toss the squash and chestnut mixture into a large mixing bowl with the egg and ricotta. Stir and press with a wooden spoon. You can pulse with an immersion blender if you desire. Taste and adjust seasoning if needed.
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.
To make the pasta dough: Combine the flour and salt on a flat work surface; shape into a mound and make a well in the center. Add the eggs and 1 tablespoon of the olive oil to the well and lightly beat with a fork. Gradually draw in the flour from the inside wall of the well in a circular motion. Use one hand for mixing and the other to protect the outer wall. Continue to incorporate all the flour until it forms a ball. Knead and fold the dough until elastic and smooth, this should take about 10 minutes. Brush the surface with the remaining olive oil and wrap the dough in plastic wrap; let rest for about 30 minutes to allow the gluten to relax.
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.
Right before assembling the lasagna, bring 5 quarts of well-salted water to a boil. Cook the pasta sheets in the water, one at a time, for 2 minutes each. Drain, cool and set aside till ready to use.