In a large pot, sauté the onions in the oil for five minutes. Add the mushrooms and sauté for another five minutes, until the mushrooms are somewhat wilted. Add the sage, 1/2 teaspoons of salt, and the wine or stock and simmer on low heat for five minutes. Set aside.
In a large bowl, stir together the eggs, pumpkin, ricotta cheese, pepper, nutmeg and remaining salt. Set aside.
Preheat the oven to 375°F Lightly grease a 9x13-inch baking dish.
Dip out about 1/2 cup of the liquid from the sautéed mushrooms and pour it into the prepared baking dish. Cover the bottom with a layer of lasagna noodles arranged close together. Evenly spread on half of the pumpkin mixture. Spoon on about a third of the sautéed mushrooms and sprinkle with 1/2 cup of ricotta salata. Add a second layer of noodles followed by the remaining pumpkin mixture, another third of the sautéed mushrooms, and 1/2 cup of ricotta salata. Finish with a third layer of noodles thoroughly moistened by the last third of the sautéed mushrooms. Evenly sprinkle on 1/2 cup of ricotta salata and top with the grated Pecorino.
Cover tightly with foil and bake for 50 minutes. Uncover and bake for an additional 10 minutes, until the lasagna is bubbly, the noodles are tender, and the top is browned. Remove from the oven and let stand for about 10 minutes before serving.