Soften noodles in a pan of very hot water while you prep the other ingredients.
Make béchamel (white sauce): Bring milk to a simmer in a saucepan and remove from heat. Melt butter in a large saucepan over medium heat. Add flour and cook, stirring, until slightly darkened, 2 minutes. Whisk milk into flour mixture all at once and whisk until smooth. Add 1 1/2 tsp. salt, 1/2 tsp. pepper, and the nutmeg. Sauce should be thick enough to coat a spoon; if it isn't, cook over medium-low heat, stirring, until thickened, 2 to 3 minutes. Remove from heat and stir in garlic, 2 tbsp. parsley, and 1/2 tbsp. thyme. Keep covered.
Preheat oven to 375°. Heat a deep, wide pot over medium-high heat 2 minutes. Swirl in 1 tbsp. oil and add leeks. Cook until tender but not browned, 3 to 4 minutes, stirring occasionally. Scoop leeks into a bowl and set aside.
Swirl 2 tbsp. oil into pot. Add mushrooms, season lightly with salt and pepper, and cook over medium heat, covered, until mushrooms are tender and beginning to release juices, about 5 minutes. Uncover and cook until edges start to brown. Stir in leeks and remaining 1/2 tbsp. thyme. Remove from heat.
Mix parmesan with asiago.
Assemble lasagna: Oil a 9- by 13-in. baking dish. Spread a few spoonfuls of béchamel over bottom. Arrange 3 noodles crosswise in dish, then spoon on about 1/2 cup béchamel, followed by a third of the mushrooms and 1/3 cup cheeses. Repeat layers twice more. Top with a final layer of noodles and béchamel, and sprinkle with remaining cheese.
Bake lasagna until browned and bubbling, about 45 minutes. Sprinkle with remaining 1 tbsp. parsley and let sit at least 15 minutes before slicing.
Make ahead: Through step 6, 1 day, chilled, or up to 3 months, frozen. Let chilled lasagna sit at room temperature 1 hour before baking. Frozen lasagna can either be thawed in the refrigerator overnight and then baked, or baked straight from the freezer for 1 3/4 hours (cover for first hour).
TIPS FOR COOKS Shop: Regardless of variety, all mushrooms should smell sweet and earthy and have dry, firm, undamaged caps. If they're spongy or sticky, steer clear. Store: Keep in a paper bag (storing them in plastic rots them), chilled, up to 4 days. Even if they become completely dry, they'll be fine in stews; the juices plump them back up. Clean: Wipe with a barely damp paper towel. If they're very dirty or sandy, swish briefly in cold water and scrub with a small brush, then dry immediately (they get soggy fast). To cook or not to cook?: Most experts advise cooking all edible mushrooms because, to varying degrees (and depending on the person), they're difficult to digest raw. Also, many have toxins that cooking destroys. However, there's no conclusive proof that eating mild raw mushrooms, especially in moderation, is harmful.