Heat olive oil in a large skillet over medium heat. Sauté onions until translucent, about 5 minutes. Add minced garlic and cook for another minute. Incorporate sliced mushrooms and cook until browned, 8-10 minutes. Season with thyme, oregano, salt, and pepper. Pour in white wine and stir until liquid evaporates, about 2-3 minutes. Fold in grape tomatoes and cook for another 2-3 minutes until softened.

In a medium saucepan, melt butter over medium heat. Whisk in flour and cook until golden, about 2-3 minutes. Gradually add warmed milk, whisking until thickened, about 5-7 minutes. Season with nutmeg, salt, and pepper, then set aside.

Preheat oven to 375°F (190°C). Cook lasagna noodles according to package instructions. In a baking dish, spread a thin layer of Bechamel sauce on the bottom. Layer 3 noodles, half mushroom mixture, spinach, more Bechamel, and a mix of cheeses. Repeat layers, finishing with last 3 noodles topped with remaining Bechamel and cheese.

Cover the assembled lasagna with aluminum foil and bake for 25 minutes. Uncover and continue baking for another 20 minutes until the top is golden and bubbly. Let rest for 10 minutes before slicing.