How to Make It
Preheat the oven to 400°. In a large bowl, toss the arugula with 2 tablespoons of the olive oil, and season with salt. Divide the arugula between 2 baking sheets and spread it evenly. Bake until crisp, 10 to 15 minutes.
Meanwhile, in a large pot of salted boiling water, cook the gnocchi according to package directions until tender. Reserve 1/2 cup of the water, and drain.
In a large skillet, melt the butter in the remaining 2 tablespoons of olive oil. Add half of the gnocchi and cook, undisturbed, until browned on the bottom, about 2 minutes. Stir in half of the lemon and half of the garlic and cook, stirring occasionally, 1 minute. Using a slotted spoon, transfer the gnocchi, lemon, and garlic to a plate. Repeat with the remaining gnocchi, lemon, and garlic.
Add the reserved pasta water to the skillet, and bring to a boil. Cook over moderately high heat until saucy, 2 to 3 minutes. Return the gnocchi mixture to the skillet, and toss to coat. Remove the skillet from the heat, add the cheese and chile, and season with salt. Transfer to a platter or plates, top with the roasted arugula, and serve.