Preheat the oven to 375 degrees F.
Cook the noodles al dente according to the directions on the package. Drain them well and spread them out onto aluminum foil or waxed paper to prevent them from sticking.
Heat the oil in a large saute pan over medium-high heat. Add the mushrooms and cook, stirring occasionally, until browned and all the liquid has evaporated, about 5 minutes. Season with 1/4 teaspoon of salt. Stir in 1 1/2 cups of the tomato sauce and simmer for 2 minutes. In a medium bowl combine the ricotta cheese, spinach, egg, the remaining 1/4 teaspoon salt, a few turns of pepper, and nutmeg.
Spread 1 cup of tomato sauce on the bottom of a 9 by 12-inch baking dish. Spread about 2 tablespoons of the ricotta mixture onto a lasagna noodle. Top with about 1 1/2 tablespoons of the mushroom mixture, roll the noodle and place it into the baking dish. Repeat with the remaining noodles. Spread the remaining 1 1/2 cups of sauce over the lasagna rolls. Top with grated cheeses, cover loosely with foil, and bake for 45 minutes. Uncover and bake for 15 minutes more.
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