Preheat the oven to 375° F. Make the Filling: Heat the tablespoon butter in a large saucepan over medium heat until melted. Add the shallots and garlic, season with salt and pepper and sauté for 2 minutes until tender. Add the spinach in batches and sauté, adding more spinach as each batch wilts down, about 6 minutes in all, until all of the spinach is added and wilted. Adjust the seasoning, transfer to a strainer over a bowl, press down with a spoon to release excess liquid, and set aside to cool slightly.
In a large bowl combine the ricotta, eggs, 1 cup Parmesan, and the basil. Season with salt and pepper, then set aside.
Make the Cheese Sauce: heat the 4 tablespoons butter in a medium saucepan over medium heat until melted. Whisk in the flour until the mixture turns a light golden color, about 3 minutes. While whisking constantly, slowly pour in the milk. Continue to cook and whisk until the mixture thickens and bubbles, about 4 minutes, adding the nutmeg, if using. Whisk in the mozzarella and Monterey Jack cheeses until they are melted, and season with salt and pepper.
If the spinach still seems wet, squeeze with your hands. Stir the drained spinach into the reserved ricotta mixture.
Lay out all of the lasagna noodles on a clean counter top. Spread the spinach-ricotta filling evenly over all of the noodles, so that each is topped with about a 1/2-inch thick layer of the spinach-ricotta mixture.
Pour a small amount of Béchamel Sauce into the bottom of a 9×13-inch pan and spread it out. Place 4 ricotta-covered noodles in the bottom of the pan to cover it in a single layer. Drizzle 1/3 of sauce over the noodles. Place another layer of the ricotta-covered noodles over the top, drizzle with another third of the sauce, and then repeat the layers once more. Sprinkle with the remaining Parmesan cheese. Bake for about 40 minutes until the top is golden and the lasagna is bubbling. If you want a more browned top, run it under the broiler for 1 or 2 minutes, watching it carefully.
Let the lasagna rest for at least 10 to 15 minutes before cutting into squares and serving warm.