Make the pasta: Combine flour, salt, and oil in the bowl of a food processor. With the motor running, slowly pour in eggs and process until dough forms a ball, about 2 minutes. Divide pasta dough into 4 equal parts, then roll out each part with a rolling pin until flat enough to fit into a hand-cranked pasta maker. Roll dough into long thin sheets. Keeping sheets covered with a damp kitchen towel while working, cut into 12 large squares. Arrange in single layers on a tray or baking sheet, using wax paper to separate layers. Cover with plastic wrap and refrigerate until ready to use.
For the pumpkin filling: Preheat oven to 350°. Bake pumpkin chunks on a lightly oiled baking sheet until tender, 30–45 minutes. Remove from oven and set aside for about 10 minutes. When cool enough to handle, remove and discard skin. Transfer pumpkin pulp to a mixing bowl, mash with a fork, then beat in egg yolks and set aside.
Heat olive oil in a large pan over medium heat. Add onions and sauté until soft, about 10 minutes. Add red pepper, amaretti, cinnamon, 2 tbsp. of the Parmigiano-Reggiano, honey, wine, and salt and cook for 3–4 minutes. Mix in pumpkin, cook for 2 minutes, then transfer to a food processor or blender and purée until smooth. Return pumpkin mixture to the same pan, cover, and keep warm over low heat.
For the sage brown butter: Heat butter in a small skillet over medium heat. Add sage and cook until butter is golden brown. Remove from heat.
Bring a pot of salted water to a boil, then reduce to a simmer. Carefully slip pasta squares into pot and cook until they rise to the surface. Use a large skimmer to remove pasta from the water. Drain well, then lay 1 square on each of 6 plates, spoon pumpkin filling onto squares, and cover with remaining pasta squares. Drizzle sage butter over each square, sprinkle with remaining Parmigiano-Reggiano, and garnish with fresh sage leaves if you like.