Heat olive oil in a large skillet over medium heat. Sauté onions for 5 minutes, add garlic for 1 minute. Add mushrooms, cook until browned (8-10 minutes). Season with thyme, oregano, salt, and pepper. Add white wine (if using), then grape tomatoes and cook for 2-3 minutes. Set aside.

Melt butter in a medium saucepan over medium heat. Whisk in flour until golden (2-3 minutes). Gradually add warmed milk, whisking until thickened (5-7 minutes). Stir in nutmeg, salt, and pepper.

Preheat oven to 375°F (190°C). Cook lasagna noodles according to package instructions until al dente (8-10 minutes), drain and lay flat.

Spread a thin layer of béchamel at the bottom of a greased baking dish. Layer noodles, half of the mushroom filling, a handful of spinach, one-third of the béchamel, and one-third of the cheeses. Repeat layers, finishing with noodles, remaining béchamel, and cheeses.

Cover with aluminum foil, bake for 25 minutes. Remove foil and bake for an additional 20 minutes or until golden and bubbly. Let rest for 10 minutes before slicing.