Adjust an oven rack to the middle position and heat the oven to 425°F. Toss the portobello mushrooms with oil, salt, and pepper on two rimmed baking sheets. Roast until the mushrooms are tender and have released their liquid, about 20 minutes, swapping the sheets halfway through.

Pulse the white mushrooms in a food processor until they are finely chopped. In a large pot over medium-high heat, cook the red onions until they are softened and just beginning to brown. Stir in the processed white mushrooms and the dried porcini. Cook until the mushrooms are deeply browned and a dark crust forms on the bottom of the pot, about 10 minutes.

Stir in most of the garlic, reserving about a teaspoon for the herb mixture. Cook until fragrant, then pour in the vermouth. Scape the bottom of the pot with a wooden spoon to release the flavorful browned bits and simmer until the liquid has completely evaporated.

Melt the butter into the mushroom mixture, then whisk in the flour until well combined. Gradually whisk in the water and milk, then bring the mixture to a simmer. Cook until the sauce is slightly thickened, about 5 minutes — it should remain relatively loose to ensure the no-boil noodles cook through. Stir in the nutmeg, salt, and pepper to taste.

Combine the basil, parsley, lemon zest, and the reserved teaspoon of raw garlic in a small bowl to create the gremolata.

Spread a thin layer of the mushroom sauce in a 13x9-inch baking dish. Arrange three noodles on top, followed by more sauce, a portion of the roasted portobellos, a sprinkle of the gremolata, and a handful of the fontina and Parmesan. Repeat this layering three more times, finishing with a final layer of noodles topped with the remaining sauce and cheese.

Cover the dish tightly with foil and bake for 25 minutes. Remove the foil and continue baking until the cheese is bubbling and the top is golden brown, about 15 minutes. Let the lasagna rest for 15 minutes to allow the layers to set before slicing.