Bring a large pot of salted water to a boil. Cook the lasagna until very al dente (1/2 of package instructions). Drain and lay flat on paper towel or clean dish cloth.
Prepare the Broccoli
Cut the broccoli into florets and steam for a few minutes until al dente. In a large skillet, over medium heat, sauté the garlic in olive oil and add the broccoli. Remove the garlic, add salt. Mix well and remove from heat. With a fork, coarsely crush the broccoli, leaving about half whole. Mix in the Parmigiano.
Prepare the Béchamel
In a saucepan, melt the butter over low heat, slowly pour in the flour and milk, constantly mixing with a wooden spoon or whisk.
Add salt and a pinch of nutmeg. Bring the mixture to a boil, mixing so that no lumps remain. Cook for a few minutes until it is a thick cream.
Prepare and Bake the Lasagna
Drizzle some olive oil over the bottom of the baking tray. Add a layer of lasagna. On top of it, distribute about 1/3 of the broccoli mixture and 1/3 of the mozzarella. Spoon about 1/4 of the béchamel evenly over the mozzarella and broccoli. Continue for 2 more layers and end with a layer of lasagna. Spoon on the remaining béchamel on top and sprinkle with Parmigiano.
Preheat the oven to 350 degrees F. Cook for about 30 minutes, until the top is golden.