How to Make It
Preheat the oven to 375°. On a rimmed baking sheet, toss the Broccolini leaves with 2 teaspoons of the olive oil and season lightly with salt. Bake for 15 to 20 minutes, until lightly browned and crisp. Let cool completely.
Meanwhile, set up a large ice water bath. In a large saucepan of salted boiling water, blanch the Broccolini until crisp-tender, 2 to 3 minutes. Using a slotted spoon, transfer to the ice water bath to cool, then drain well and pat dry.
In the saucepan of salted boiling water, cook the spaghetti until al dente. Reserve 1 cup of the pasta water. Drain the pasta and toss with 2 teaspoons of the olive oil.
Wipe out the saucepan and heat the remaining 2 tablespoons of olive oil in it. Add the garlic and cook over moderate heat, stirring, until softened, about 1 minute. Add the Broccolini and cook over moderately high heat, stirring frequently, until the garlic is golden, about 3 minutes. Add the spaghetti and the reserved pasta water and cook, tossing, until coated in a light sauce, 2 to 3 minutes. Remove from the heat. While tossing, add the butter 1 tablespoon at a time until incorporated. Stir in the lemon juice and season generously with salt and pepper. Transfer to bowls. Top with the crispy Broccolini leaves and more pepper and serve right away, passing Parmesan at the table.