Place both halves of the squash, face down, on a baking sheet and pour a little water into a pan. Bake in a 350°F (180°C) oven for 30 to 40 minutes or until the squash is fork tender. Remove from the oven.
Using a spoon, begin scraping out the flesh of the squash from the skin and place in a food processor. Add butter, salt and honey, blend until smooth. Place in a small pot to stay warm.
Bring a large pot of salted water to a boil. Add pasta and cook, stirring occasionally, until al dente, about 4 minutes. Drain pasta, reserving 1/2 cup (125 mL) pasta water. Set aside separately.
Melt butter in a 12-inch (30 cm) skillet over medium heat. Add the pine nuts and cook, stirring often until golden brown, about 4 to 5 minutes. Using a slotted spoon, transfer pine nuts to a bowl.
In the same pan sauté shallots and hen of woods mushrooms, until caramelized, about 4 minutes. Remove from pan.
Add pasta, hen of woods mushrooms and half the pine nuts to the skillet and toss until hot. Stir in reserved pasta water to create a sauce, and season with salt and pepper. Finally, add sage and chives. Set aside.
Preheat another skillet over medium heat with 2 tbsp of butter. Working in two batches, crack eggs into butter, and cook, spooning butter over the yolks, until whites are set but the yolks are still runny, about 3 minutes. Transfer eggs to a plate and keep warm.
Spoon butternut squash puree evenly on bottom of serving bowls.
Divide pasta on top of squash and top each with an egg. Sprinkle remaining pine nuts, Parmesan and nutmeg.
Pile flour and semolina into a large bowl and make a well in the middle.
In a small bowl, add yolks, oil, 4 tbsp (60 mL) of cold water and mix together then add to the well. Gradually bring the flour in from the outside. Bring it together into a ball of dough.
Knead on a flour-dusted surface until smooth and elastic, about 5 minutes. Wrap in plastic wrap and leave to relax for 30 minutes.
Roll out pasta using a pasta roller then cut into thin (approximately 1 inch) ribbons.