Preheat the oven to 400 degrees F.
Coat a 9x13-inch baking dish with nonstick spray.
In a large skillet over medium heat, add 1 1/2 tablespoons of the olive oil.
Add the Brussels, the shallots, the salt, and the pepper to the oil and stir to coat.
Cook the mixture, stirring occasionally, until the Brussels have softened and become bright in color, about 5-6 minutes.
Stir the balsamic vinegar into the Brussels mixture and cook for 1 minute.
Transfer the Brussels mixture into a bowl.
In the same skillet, add the remaining olive oil.
Add the cremini mushrooms, the shiitake mushrooms, and the garlic to the oil and stir well to coat.
Cook the mushroom mixture, stirring occasionally, until the mushrooms are soft and juicy, about 6-8 minutes.
Turn off the heat and season the mushroom mixture with the salt and the pepper.
In a saucepan over medium heat, heat the butter until sizzling.
Whisk the flour into the butter until a roux forms.
Cook the roux until it becomes golden and fragrant, about 2-3 minutes.
Slowly stream the milk into the roux while whisking constantly to ensure the mixture comes together.
Cook the sauce mixture, stirring constantly, until it starts to thicken, about 5 minutes.
Season the sauce mixture with the salt, the pepper, and the nutmeg.
In the prepared dish, add 1/4 cup of the sauce to the bottom.
Add a layer of the lasagna noodles, top with 1/3 of the Brussels, and then 1/3 of the mushrooms.
On top of the mushrooms, sprinkle 1/3 cup of the provolone cheese, then 1/4 cup of the ricotta.
Pour 1/3 of the sauce on top of the cheese.
Repeat the layers two more times starting with the lasagna noodles.
Top the lasagna with any of the remaining provolone cheese, the Pecorino Ramano cheese, and the remaining sauce.
Bake until the top of the lasagna is golden and bubbly, about 25-30 minutes.
Let the lasagna sit for 20 minutes.
Slice and serve.