Heat olive oil in a large skillet over medium-high heat. Add the shallots and cook for 2 minutes until softened. Add the mushrooms and cook for 8 to 10 minutes, stirring occasionally, until deeply browned and all released liquid has evaporated. Stir in the garlic and baby spinach and cook until the spinach is just wilted, about 1 to 2 minutes. Season with salt and pepper and set aside.

In a medium saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1 to 2 minutes, whisking constantly, until the mixture is smooth and smells slightly nutty. Gradually add the milk, about half a cup at a time, whisking well after each addition. Bring to a gentle simmer and whisk until the sauce thickens enough to coat the back of a spoon, about 5 minutes. Remove from heat and stir in the 1/2 cup shredded mozzarella, nutmeg, salt, and pepper.

Preheat the oven to 400°F (200°C). Grease a 9x13-inch baking dish. Give the ricotta a quick stir to loosen its texture.

Spread 1/2 cup of béchamel across the bottom of the prepared baking dish. Add a layer of no-boil lasagna noodles. Dollop half the ricotta across the noodles, scatter half the mushroom and spinach filling over the top, drizzle with béchamel, and finish with a sprinkle of Parmesan and mozzarella.

Repeat the layering process with the remaining ricotta, mushroom filling, béchamel, and cheese. Top with a final layer of noodles. Pour the remaining béchamel over the top, spreading it to the corners and edges. Scatter the remaining mozzarella and Parmesan over the surface.

Cover the baking dish tightly with foil and bake for 40 minutes.

Remove the foil and bake for another 10 minutes until the top is golden and bubbling. For deeper browning, broil for 2 minutes with the oven rack in the upper third, watching closely.

Remove from the oven and let the lasagna rest for 15 minutes before slicing. This allows the layers to firm up for clean, defined slices.