Prepare the broccoli: Break the broccoli into small florets by hand and chop the stem finely. Boil in salted water with lid on for about 7 minutes until tender. Drain and set aside.
Cook the mushroom and meat filling: Heat extra virgin olive oil and a splash of water in a pan over medium-high heat. Sauté chopped onion for 5 minutes. Add a splash of wine, then sliced mushrooms, butter, and chopped parsley. Cook until mushrooms release water, then add ground beef, breaking it up and cooking until browned. Season with salt and pepper. Add remaining wine and simmer until liquid evaporates.
Combine broccoli with meat mixture: Add boiled broccoli to the meat and mushroom mixture. Lightly mash broccoli to blend but not puree. Add torn basil leaves, stir, and turn off heat.
Make the béchamel sauce: In a small pot over medium-low heat, melt butter. Stir in flour and cook briefly to form a roux. Gradually whisk in milk in small amounts, stirring constantly to avoid lumps. Continue adding milk until sauce is smooth and creamy. Add nutmeg, stir, and remove from heat.
Assemble the lasagna: Spread a layer of béchamel sauce on the bottom of an 11x7.5x3 inch Pyrex dish. Add a small amount of meat mixture. Place lasagna sheets to cover, breaking to fill gaps. Spread béchamel over sheets, then add a generous layer of meat mixture, followed by provolone, mozzarella, and pecorino cheeses. Repeat layering until halfway, pressing down to compact. Continue layering remaining ingredients, closing gaps and pressing down firmly. Finish with a top layer of béchamel sauce.
Bake the lasagna covered: Cover the dish with foil, ensuring foil does not touch the béchamel surface by leaving a gap. Bake in a preheated oven at 180°C (356°F) for 20 minutes.
Add cheese topping and finish baking: Remove foil. Top the lasagna with provolone, mozzarella, and pecorino cheeses, and drizzle with extra virgin olive oil. Bake uncovered at 180°C (356°F) for about 10 minutes or until cheese melts and forms a golden crust.
Rest and serve: Allow the lasagna to rest for about 20 minutes to set and keep shape before slicing and serving.