Season the chicken breast evenly with sea salt and black pepper on both sides.
Heat a non-stick skillet over medium-high heat and sear the chicken for 5-6 minutes per side until golden brown and cooked through.
Remove the chicken from the pan, let it rest for 3 minutes, then slice into bite-sized strips.
Bring a medium pot of salted water to a boil and cook the potato gnocchi until they float to the surface, about 2-3 minutes.
While gnocchi cooks, place the arugula, walnuts, garlic, lemon juice, olive oil, and parmesan in a food processor.
Pulse the pesto ingredients until a slightly textured, vibrant green sauce forms.
Drain the gnocchi, reserving 1 tablespoon of the cooking water.
In a large bowl, toss the warm gnocchi, sliced chicken, and arugula pesto together, adding the reserved water if needed to loosen the sauce.