In a large skillet, heat olive oil over medium heat. Add sliced mushrooms and sauté for 5-7 minutes until golden brown. Toss in fresh spinach, cooking until just wilted, about 2-3 minutes.

In a saucepan, melt unsalted butter over medium heat. Sprinkle in gluten-free flour, whisking for about 2 minutes to create a roux. Gradually add whole milk, whisking until the sauce thickens (5-7 minutes). Season with salt and nutmeg.

Preheat oven to 350°F (175°C). In a 9x13-inch baking dish, spread a thin layer of béchamel sauce on the bottom. Layer lasagna noodles, half of the mushroom and spinach mixture, a third of the béchamel sauce, and cheese. Repeat layers and finish with noodles, béchamel, and cheese.

Cover with aluminum foil and bake for 30 minutes. Remove foil and bake for an additional 10-15 minutes until bubbly and golden.

Remove from oven and let rest for 10-15 minutes before serving.