Blanch the basil in boiling water for 20 seconds. Drain and rinse under cold water; squeeze dry. In a mini food processor, puree the basil with 1/2 cup of the oil. Season with salt and transfer to a small bowl.
In a large skillet, heat 2 tablespoons of the oil. Add the garlic and cook for 30 seconds. Add the broccoli, 1/2 cup of water and the butter, cover and cook over moderately low heat until tender, 10 minutes. Let cool. Coarsely puree the broccoli in a food processor; transfer to a bowl. Stir in the ricotta and 1/4 cup of the pecorino and season with salt.
Bring a large pot of salted water to a boil. Using a pasta machine, roll the lasagna sheets through the thinnest setting. Cut the sheets into twelve 4-inch squares. Cook the squares in the water until tender, 2 minutes. Drain, pat dry and rub with oil to prevent sticking.
Preheat the oven to 425°. Coat a large, shallow baking dish with oil. Place 2 heaping tablespoons of the filling into the center of each pasta square. Fold the squares into triangles; arrange in the dish. Sprinkle with 3 tablespoons of the pecorino. Bake on the top rack for 10 minutes, until the cheese starts to brown. Drizzle with the basil oil and serve, passing more cheese at the table.