Prepare the lasagna noodles: Bring a large pot of lightly salted water to a boil and fill a large mixing bowl with ice and water. Boil the noodles until al dente, tender but still very chewy, about 8 minutes. Drain them and immediately transfer to the bowl of ice water.
Lightly oil a baking sheet with some olive oil. Drain the noodles from the ice water and arrange them in a single layer on the baking sheet, not overlapping. Set aside.
While the noodles are cooking, start the Bechamel Sauce: In a large saucepan over low heat, melt the butter. Stirring continuously with a wire whisk, sprinkle in the flour; continue cooking, stirring constantly, for 5 minutes. Whisking briskly, slowly pour in the milk. Stir in the goat cheese, salt and pinch of nutmeg. Bring to a gentle simmer and continue cooking, stirring frequently, until thick and creamy, about 10 minutes. Remove from the heat and set aside.
To make the Filling: In a large mixing bowl, combine the pumpkin puree, thyme, sage, nutmeg, 1/2 cup breadcrumbs, mascarpone, 1/4 cup Parmesan, brown sugar, goat cheese, salt and pepper. Stir with a wooden spoon until thoroughly combined.
Preheat the oven to 400 degrees.
For the Lasagna Assembly: Lightly coat the inside of a 13-by-9-by-3-inch lasagna pan with nonstick cooking spray. Spread a third of the pumpkin mixture on the bottom of the pan. Cover with a layer of lasagna noodles, placed lengthwise side by side. Spread another third of the filling over the noodles, and top with 3/4 cup of Bechamel. Top with another layer of pasta, the remaining filling, and another 3/4 cup of Bechamel. Cover with a final layer of noodles and all the remaining Bechamel. Evenly sprinkle with the 1/2 cup each of breadcrumbs and Parmesan.
Spray a sheet of aluminum foil with nonstick spray and place it sprayed side down over the lasagna pan, folding the edges over.
Bake until the lasagna is heated through and bubbling, about 40 minutes. Remove the foil and bake until the topping is golden brown, about 10 minutes.
Remove the pan from the oven and let the lasagna settle for 10 minutes. With a sharp knife, cut it into rectangles or squares. Use a spatula to transfer them from the pan to serving plates.