Preheat the oven to 375°F (190°C).
Melt the unsalted butter in a large saucepan set over medium heat.
Add the finely diced onion and minced garlic. Cook for approximately 7 minutes, stirring from time to time, until the onion has softened and the mixture is fragrant.
Pour the apple cider vinegar into the saucepan. Continue cooking and stirring for about 4 minutes, or until most of the liquid has evaporated.
Mix in the dried sage, followed by the pumpkin purée, maple syrup, and freshly grated nutmeg.
Season the pumpkin mixture with kosher salt and freshly ground black pepper.
Cook for an additional 5 minutes, stirring frequently, until the sauce is evenly heated and the seasonings are well incorporated. Taste and adjust the salt and pepper as necessary.
Place the beaten egg, ricotta, 2 cups of mozzarella, and 2 cups of Fontina in a medium mixing bowl.
Stir thoroughly until the cheese filling is smooth and evenly combined.
Lightly coat a 13 × 9-inch baking dish with nonstick cooking spray.
Spread a thin coating of the prepared pumpkin sauce across the base of the baking dish.
Arrange approximately one-quarter of the oven-ready lasagna noodles over the sauce.
Distribute one-third of the cheese filling evenly across the noodles.
Spoon one-third of the remaining pumpkin sauce over the cheese layer, then add another portion of noodles and another third of the cheese filling.
Repeat the layering process once more with noodles, cheese filling, and pumpkin sauce.
Complete the lasagna with the remaining pumpkin mixture and a final layer of noodles.
Scatter 1 cup of shredded Fontina and 1 cup of shredded mozzarella evenly over the assembled lasagna.
Cover the baking dish securely with aluminum foil and bake at 375°F (190°C) for 35 minutes.
Remove the foil and raise the oven temperature to 400°F (205°C).
Return the uncovered lasagna to the oven and bake for approximately 20 minutes, until the cheese has melted and developed a lightly browned surface.
Remove the dish from the oven and allow the lasagna to stand for 5 minutes before cutting and serving.