Bring a medium pot of salted water to a boil for the gnocchi.
Season the chicken breast evenly 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. Remove from heat, let rest for 5 minutes, and slice into strips.
While the chicken cooks, prepare the pesto by combining baby arugula, walnut halves, garlic, lemon juice, parmesan cheese, and olive oil in a small food processor.
Pulse the pesto ingredients until a smooth, bright green consistency is achieved, adding a teaspoon of water if needed to thin it out.
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 1 tablespoon of the cooking water.
In the same skillet used for the chicken, combine the cooked gnocchi, arugula pesto, sliced chicken, and reserved cooking water.
Toss everything gently over low heat for 1 minute until the sauce is creamy and coats the gnocchi evenly before serving.