Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium heat and sear the chicken for 6-7 minutes per side until fully cooked, then set aside to rest.
Bring a pot of salted water to a boil and cook the gnocchi until they float to the surface, approximately 2-3 minutes, then drain.
In a food processor, combine the arugula, walnuts, garlic, parmesan, lemon juice, olive oil, and the remaining salt and pepper.
Pulse the mixture until a smooth, thick pesto forms, adding a teaspoon of water if needed to reach your desired consistency.
Slice the rested chicken into thin strips.
Toss the warm gnocchi and sliced chicken with the arugula pesto in a large bowl until every piece is thoroughly coated.
Transfer to a plate and serve immediately.