Place 4 cups arugula, 1 cup Parmesan cheese, walnuts, salt, and pepper in a food processor and pulse until incorporated, 30 seconds. With motor running, drizzle in olive oil and water, and process until smooth, another 30 seconds, adding more water by the tablespoonful if necessary.
Cook gnocchi according to package directions; drain. Return gnocchi to pot and add pesto.
Divide remaining 2 cups arugula among 4 bowls and toss with the gnocchi.
Sprinkle with remaining Parmesan and serve immediately.
