Cover porcini with water in microwave-safe dish; cover with plastic wrap, cut several steam vents in plastic with paring knife, and microwave on high power for 30 seconds. Let stand until mushrooms soften, about 5 minutes. Lift mushrooms from liquid with fork and roughly chop. Strain liquid though fine-mesh strainer lined with paper towels into medium bowl. Set mushrooms and liquid aside.
Adjust oven rack to middle position and heat oven to 425 degrees.
Spread portobellos in even layer on rimmed baking sheet and drizzle with 2 tbsp oil; tossing to coat mushrooms evenly; sprinkle with 1/2 teaspoons salt and 1/2 tsp pepper and toss again. Roast mushrooms until shriveled and all liquid has evaporated, about 30 minutes. Set aside to cool. Do not turn off oven.
Heat 1 tablespoons oil in 12 inch nonstick skillet over medium high heat until shimmering. Add onions, 1/4 teaspoons salt and 1/4 teaspoons of pepper and cook, stirring occasionally, until onions are browned around the edges, about 10 minutes. Transfer onions to large bowl and set aside.
Process button mushrooms in food processor until uniformly coarsely chopped, about six 1-second pulses, stopping to scrape bowl as needed. Heat remaining 1 tablespoons oil in no-empty skillet over medium high heat until shimmering. Add chopped button mushrooms and cook, stirring occasionally, until browned and moisture has evaporated, 6-8 minutes.
Reduce heat to medium and stir in porcini mushrooms, 1 tablespoons garlic, 1 teaspoons salt, and 1 teaspoons pepper. Cook, stirring frequently, until garlic is fragrant, about 1 minute. Add vermouth and cook, stirring occasionally, until liquid has evaporated, 2-3 minutes.
Add butter and cook until melted. Add flour and cook, stirring constantly, about 1 minute. Add milk, scraping pan bottom to loosen brown bits. Add reserve porcini liquid and nutmeg. Increase heat to medium high and bring mixture to boil. Reduce heat to medium low and simmer until sauce reaches consistency of heavy cream, 10-15 minutes. Remove from heat and stir in 2 tablespoons parsley and 1/4 cup basil.
Combine fontina and Parmesan in medium bowl. Toss cooled portabello mushrooms with onions in large bowl. Place noodles in 13 X 9 inch ovensafe baking dish and cover with hot tap water; let soak 5 minutes, stirring occasionally to prevent sticking. Remove noodles from water and place in single layer on kitchen towel. Wipe baking dish dry and coat with butter.
Using rubber spatula, evenly distribute 1 cup mushroom sauce in bottom of baking dish; position 3 noodles on top of sauce. Spread 3/4 cup mushroom sauce evenly over noodles followed by 2 cups mushroom-onion mixture and 3/4 cup cheese. Repeat layering of noodles, sauce, mushroom-onion mixture, and cheese two more times. Place 3 remaining noodles on top of last layer of cheese. Spread remaining sauce over noodles and sprinkle with remainin cheese. Lightly spray laarge sheet of foil with nonstick cooking spray and cover lasagna. Bake until bubbling, about 20 minutes.
While lasagna is baking, combine remaining 2 tablespoons parsley, 2 tablespoons basil and 1 tsp garlic with zest in small bowl. Increase oven temperature to 500 degrees, remove foil from lasagna, and continue to bake until cheese on top becomes spotty brown, 6-8 minutes. Remove lasagna from oven and immediately sprinkle evenly with herb mixture. Cool for 15 minutes, then cut into pieces and serve.