Heat oven to 425 degrees F. Toss the portobello mushrooms with 2 tablespoons olive oil, ½ teaspoon salt and ½ teaspoon pepper, and spread out on a rimmed baking sheet. Roast the mushrooms until shriveled and all liquid released from the mushrooms has evaporated, about 30 minutes, stirring halfway through roasting. Transfer mushrooms to a large bowl and set aside.
Meanwhile, microwave the water and porcini mushrooms in a covered bowl until steaming, about 1 minute. Let sit until the mushrooms are softened, about 5 minutes. Drain the mushrooms through a fine-mesh sieve lined with a coffee filter and set over a bowl. Reserve the liquid and coarsely chop the mushrooms.
Heat 1 tablespoon of the olive oil in a 12-inch skillet over medium heat until shimmering. Add the onions, ¼ teaspoon salt, ¼ teaspoon pepper, and cook, stirring occasionally until softened and lightly browned, 8 to 10 minutes. Transfer the onions to the bowl with the roasted mushrooms.
Heat the remaining 1 tablespoon olive oil in the empty skillet over medium-high heat until shimmering. Add the minced white mushrooms and cook, stirring occasionally, until browned and all moisture has evaporated, 6 to 8 minutes. Stir in the chopped porcini mushrooms, 3 of the minced garlic cloves, 1 teaspoon salt and 1 teaspoon pepper; reduce heat to medium; and cook, stirring often, until the garlic is fragrant, about 30 seconds. Stir in the vermouth and cook, stirring occasionally, until the liquid has evaporated, 2 to 3 minutes.
Stir in the butter and allow to melt. Add the flour and cook, stirring constantly, for 1 minute. Slowly stir in reserved porcini mushroom liquid, scraping up any browned bits. Gradually add the milk, stirring constantly, then stir in the nutmeg and bring the sauce to a simmer. Cook until the sauce has thickened and measures about 4 cups, 10 to 15 minutes. Off the heat, stir in ¼ cup of the basil and 2 tablespoons of the parsley.
Pour 2 inches of boiling water into a 9x13-inch baking dish. Slip the lasagna noodles into the water, one at a time, and soak until pliable, about 5 minutes, separating noodles with the tip of a sharp knife to prevent sticking. Remove the noodles from the water and place in a single layer on clean dish towels; discard the water. Dry the dish and spray with non-stick cooking spray.
Toss together the shredded Fontina and grated Parmesan cheeses in a small bowl. Spread 1 cup of the mushroom sauce evenly over the bottom of the prepared dish. Arrange 3 noodles in a single layer on top of the sauce. Spread ¾ cup of the sauce evenly over the noodles, then sprinkle with 2 cups of the mushroom-onion mixture and ¾ cup of the cheese mixture. Repeat the layering of noodles, sauce, mushroom-onion mixture, and cheese 2 more times. For the final layer, arrange the remaining 3 noodles on top and cover completely with the remaining sauce. Sprinkle with the remaining cheese.
Cover the dish tightly with aluminum foil that has been sprayed with non-stick cooking spray and bake until the edges are just bubbling, about 20 minutes, rotating the dish halfway through baking. While the lasagna is baking, combine the remaining 1 clove minced garlic, remaining 2 tablespoons basil, and remaining 2 tablespoons parsley in a small bowl.
Remove the foil, increase the oven temperature to 500 degrees F, and continue to bake until the cheese on top becomes spotty brown, 6 to 8 minutes longer. Sprinkle the lasagna with the garlic-herb mixture and let cool for 15 minutes before serving.