Preheat oven to 375°.
Combine the water and porcini in a medium bowl; cover and let stand for 15 minutes. Drain in a sieve over a bowl, reserving 1 cup soaking liquid. Bring the soaking liquid to a boil in small saucepan; reduce heat to medium, and cook until reduced to 1/4 cup (about 5 minutes). Remove the stems from shiitakes; discard stems. Slice shiitake caps.
Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add porcini, shiitake, and button mushrooms, 1/4 teaspoon salt, thyme, and 1/8 teaspoon pepper; sauté 12 minutes or until softened. Add wine; cook 2 minutes or until liquid almost evaporates.
Melt the butter in a medium saucepan over medium heat. Add the flour, and cook for 30 seconds, stirring constantly with a whisk. Gradually add 1/4 cup porcini mushroom liquid and milk, stirring with a whisk until blended. Cook until mixture thickens (about 4 minutes). Remove from heat. Add the goat cheese, 2 tablespoons Parmesan, 1/4 teaspoon salt, and1/8 teaspoon pepper, and stir until cheese melts.
Spread 1/2 cup cheese sauce in bottom of an 8-inch square baking dish coated with cooking spray. Arrange 2 noodles over cheese sauce, and top with half of mushroom mixture, half of spinach, 3/4 cup tomato, 1/2 cup cheese sauce, and 1/4 cup mozzarella. Repeat layers, ending with noodles. Spread remaining cheese sauce over noodles. Sprinkle with 1/4 cup mozzarella, 1/2 cup tomato, and 2 tablespoons Parmesan. Cover and bake at 375° for 25 minutes. Uncover and bake an additional 25 minutes. Let stand 5 minutes.