Heat the olive oil in a large skillet over medium heat. Add the chopped onions and cook until translucent, about 5 minutes. Add minced garlic and cook for an additional minute.

Add sliced mushrooms to the skillet. Cook until mushrooms release moisture and begin to brown, about 8–10 minutes. Season with thyme, oregano, salt, and pepper. If using, pour in white wine and cook until evaporated, about 2–3 minutes. Add halved grape tomatoes and cook for an additional 2–3 minutes until softened. Set aside.

In a medium saucepan, melt butter over medium heat. Add flour and whisk until light golden, about 2–3 minutes. Gradually add warmed milk, whisking constantly until thickened, about 5–7 minutes. Season with nutmeg, salt, and pepper.

Preheat oven to 375°F (190°C). Cook lasagna noodles according to package instructions until al dente. Drain and lay flat on a clean kitchen towel.

Spread a thin layer of Bechamel sauce on the bottom of a 9×13 inch baking dish. Layer noodles, half of the mushroom and tomato mixture, half of the chopped spinach, one-third of the Bechamel sauce, and one-third of the cheeses. Repeat layers. Finish with a final layer of noodles, remaining Bechamel sauce, and rest of the cheeses.

Cover with aluminum foil and bake for 25 minutes. Remove foil and bake for an additional 20 minutes until golden and bubbly. Let rest for 10 minutes before serving. Garnish with fresh basil if desired.