To prepare filling, combine water and porcini in a small bowl; let stand 30 minutes. Drain porcini through a sieve over a bowl, squeezing porcini to extract liquid. Reserve 1 1/2 cups liquid; discard remaining liquid. Rinse porcini, and drain. Chop coarsely; set aside.
Heat oil in a large Dutch oven over medium-high heat. Add onion; sauté 5 minutes or until tender. Add garlic; sauté 30 seconds. Add shiitake mushrooms, cremini mushrooms, and 1/2 teaspoon salt. Sauté 5 minutes or until mushrooms release moisture and begin to brown. Stir in porcini, wine, and soy sauce. Cook 4 minutes or until liquid almost evaporates, stirring frequently. Stir in parsley and rosemary. Add the reserved porcini liquid; bring to a simmer. Cook until liquid is reduced to 3/4 cup (about 10 minutes). Remove from heat; stir in 1/4 teaspoon pepper.
To prepare béchamel, place milk in a 4-cup glass measure. Microwave at HIGH 3 minutes or until hot, stirring after 2 minutes. Melt butter in a large, heavy saucepan over medium heat. Add flour, stirring with a whisk, and cook 1 minute or until bubbly, stirring constantly. Gradually add milk, stirring constantly with a whisk. Bring to a simmer; cook over low heat 8 minutes or until sauce slightly thickens, stirring frequently. Remove from heat, and stir in 1/2 teaspoon salt, 1/4 teaspoon pepper, and nutmeg.
Preheat oven to 350°.
Spread 1/2 cup béchamel in bottom of an 11 x 7-inch baking dish coated with cooking spray. Arrange 3 noodles over béchamel; top with 1/2 cup béchamel. Top with 1 cup mushroom mixture. Sprinkle with 1/4 cup cheese. Repeat layers three times (dish will be very full). Cover with foil; place baking dish on a baking sheet. Bake at 350° for 30 minutes. Uncover lasagna; bake an additional 15 minutes or until cheese melts.