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 with 0.25 tablespoons of the olive oil and sear the chicken until golden and cooked through, then slice into thin strips.
In a small food processor, combine the baby arugula, walnuts, garlic, lemon juice, parmesan cheese, and the remaining 0.5 tablespoons of olive oil.
Pulse the mixture until a slightly textured pesto forms.
Add the potato gnocchi to the boiling water and cook until they float to the surface, then drain, reserving 2 tablespoons of the cooking water.
Return the gnocchi to the pot over low heat, adding the arugula pesto, sliced chicken, and reserved water.
Toss gently until the gnocchi is evenly coated and fragrant, then finish with the remaining sea salt and black pepper.