In the bowl of a food processor, combine arugula, basil, pine nuts, Parmesan, and garlic. With the motor running, add 1 cup olive oil in a slow stream until emulsified. Season with salt and pepper, to taste.
In a large pot of boiling salted water, add gnocchi. Cook until they rise to the surface (just a few minutes for fresh pasta.) Drain well, reserving 1/4 cup of the pasta water.
In a large pan or bowl, combine pesto with remaining 1/4 cup olive oil and reserved pasta water. Pesto should become a sauce-like consistency. Add gnocchi and toss to combine. Serve, garnished with extra cheese.