Adjust the oven rack to the middle position and heat the oven to 425°F. Toss the portobello mushrooms with oil, salt, and pepper on a rimmed baking sheet and roast until shriveled and browned, about 30 minutes, stirring halfway through.

Heat oil in a large nonstick skillet over medium heat until shimmering. Add the onions, salt, and pepper and cook until softened and lightly browned, 8 to 10 minutes, then transfer to the bowl with the roasted portobellos.

Pulse the white mushrooms in a food processor until coarsely chopped. Heat oil in the empty skillet over medium-high heat and cook the chopped mushrooms and porcini until browned and the moisture has completely evaporated, 6 to 8 minutes. Stir in most of the garlic, salt, and pepper and cook until fragrant; then pour in the vermouth and cook until the liquid has evaporated, 2 to 3 minutes.

Melt the butter in the skillet, whisk in the flour, and cook for 1 minute. Slowly whisk in the water, milk, and nutmeg and simmer until the sauce thickens enough to coat a spoon, 10 to 15 minutes. Off the heat, stir in a portion of the basil and parsley.

Pour boiling water into a 13 by 9-inch baking dish and soak the noodles until pliable, about 5 minutes, using a knife to keep them from sticking. Remove the noodles to clean dish towels in a single layer, then dry and grease the baking dish.

Combine the fontina and Parmesan. Spread a layer of mushroom sauce over the bottom of the dish, top with 3 noodles, more sauce, the mushroom-onion mixture, and a portion of the cheese. Repeat the layers twice more, then finish with the final 3 noodles, the remaining sauce, and the rest of the cheese.

Cover the dish tightly with greased foil and bake until the edges are bubbling, about 20 minutes. Remove the foil, increase the oven temperature to 500°F, and bake until the cheese is spotty brown, 6 to 8 minutes.

Mix the remaining garlic, parsley, basil, and lemon zest together and sprinkle the gremolata over the hot lasagna. Let the casserole rest for 15 minutes to set the layers before serving.