Bring a medium pot of salted water to a boil.
Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium-high heat and sear the chicken for 6-7 minutes per side until cooked through, then set aside to rest before slicing into thin strips.
In a food processor, combine the baby arugula, walnuts, garlic, lemon juice, parmesan cheese, and extra virgin olive oil.
Pulse the mixture until a slightly textured pesto forms, adding a teaspoon of water if needed to reach a smooth consistency.
Add the potato gnocchi to the boiling water and cook for 2-3 minutes, or until they float to the surface.
Drain the gnocchi, reserving two tablespoons of the cooking water.
In a large bowl, toss the warm gnocchi and sliced chicken with the arugula pesto and reserved pasta water until well coated.
Season with the remaining sea salt and black pepper before serving immediately.