Bring a medium pot of salted water to a boil and cook the potato gnocchi according to package instructions until they float, then drain and set aside.
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 5-6 minutes per side until fully cooked, then slice into thin strips.
In a small food processor, combine the arugula, walnuts, parmesan cheese, garlic, lemon juice, and the remaining sea salt and pepper.
Pulse the pesto ingredients while slowly drizzling in the olive oil until a slightly textured paste forms.
In a large bowl, toss the cooked gnocchi and sliced chicken with the fresh arugula pesto until evenly coated.
Serve immediately while warm, garnished with an extra pinch of black pepper if desired.