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 the internal temperature reaches 165°F.
While the chicken is cooking, prepare the pesto by placing the arugula, walnuts, garlic, lemon juice, olive oil, water, and remaining salt and pepper into a small food processor.
Pulse the pesto ingredients until smooth, adding an extra teaspoon of water if needed to reach your desired consistency.
Add the gnocchi to the boiling water and cook until they float to the surface, which usually takes about 2 to 3 minutes.
Drain the gnocchi and return them to the pot, then toss immediately with the fresh arugula pesto.
Slice the cooked chicken breast into thin strips and serve it over the warm, pesto-coated gnocchi.