Preheat the oven to 375 degrees F.
Bring a few inches water to a boil in a large pot and add salt. Cook the escarole for 5 minutes, then drain.
Cover the garlic cloves with water in a saucepan and simmer for 20 minutes. Drain and let cool. Squish the garlic pulp from the jackets into a bowl and mash with a fork.
Meanwhile, melt 3 tablespoons butter in a medium saucepan over medium heat. Add the sage and stir. Add the squash, stir and season with salt and pepper. Add the chicken stock and cook, stirring occasionally, for 15 minutes. All the stock should be absorbed and the squash should be lightly browned and tender. Remove from the heat.
Melt the remaining 3 tablespoons butter in another medium saucepan over medium heat. Add the flour and whisk for 1 minute. Whisk in the milk and mashed garlic and season with salt, pepper and a little nutmeg. Cook until slightly thickened (the sauce should just coat the back of the spoon evenly without breaking up on the spoon's surface).
Whisk together the pumpkin puree, 2 eggs and some salt and pepper in a small bowl.
In another bowl, whisk together the ricotta, half the Parmigiano-Reggiano and the remaining egg.
Preheat the oven to 375 degrees F. Soak the lasagna sheets in water for 5 minutes.
Pour about half the garlic sauce into the bottom of a 9 by 11-inch baking dish. Add a layer of lasagna sheets and then half the pumpkin mixture. Make another layer of lasagna sheets and spread with all of the ricotta mixture, followed by the squash and then escarole. Top with half the Fontina, another layer of lasagna sheets, the remaining pumpkin and another layer of lasagna sheets. Finish with the remaining garlic sauce and sprinkle with the remaining Parmigiano-Reggiano and Fontina.
Bake the lasagna, covered, for 45 minutes. Raise the oven temperature to 400 degrees F and continue to bake, uncovered, for 15 minutes more. Let rest for 15 minutes, then cut and serve.
The lasagna can be wrapped tightly and refrigerated or frozen for a make-ahead meal. Reheat, covered, until hot, 45 minutes to 1 hour, then uncover and bake for 10 to 15 minutes more.