Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium-high heat and cook the chicken for 6-7 minutes per side until the internal temperature reaches 165°F, then let it rest for 5 minutes before slicing into thin strips.
While the chicken cooks, prepare the pesto by combining the baby arugula, walnuts, extra virgin olive oil, parmesan cheese, garlic, lemon juice, and water in a small food processor.
Pulse the pesto ingredients until smooth and well combined, adding an extra teaspoon of water if needed to reach a creamy consistency.
Bring a medium pot of salted water to a boil and cook the potato gnocchi according to the package instructions, usually about 2-3 minutes until they float to the surface.
Drain the gnocchi and return them to the warm pot, then gently fold in the arugula pesto and the sliced chicken strips.
Season the final dish with the remaining sea salt and black pepper and serve immediately while warm.