Bring a large pot of water to a boil.
For the pesto: In a food processor, combine the basil, parsley, cheese, pine nuts, garlic and lemon zest and blend. With the motor running, stream in the olive oil. Season with salt and pepper.
For the gnocchi with pole beans: Blanch the beans briefly in the boiling water. Remove with a slotted spoon and set aside. Add the gnocchi to the boiling water and cook to al dente according to the package directions. Drain.
In a large nonstick pan, heat the olive oil over medium-high heat. Add the shallots and gnocchi and saute until golden brown. Add the beans and turn the heat down to medium. Add the butter, lemon zest and juice and 2 tablespoons of the pesto. Cook until the beans are just tender. Season as needed.
Divide the gnocchi among 4 dishes and top with the pea greens, a drizzle of extra-virgin olive oil and a sprinkling of flaky sea salt and Parmigiano-Reggiano.
You will have more pesto than needed for this recipe. Cover the excess with olive oil and place it in the fridge for up to 4 days; it also freezes well.