Bring a small pot of salted water to a boil and cook the potato gnocchi according to package directions until they float, then drain and set aside.
Season the chicken breast with half of the sea salt and half of the black pepper.
Heat a non-stick skillet over medium-high heat and sear the chicken for 5-7 minutes per side until golden and cooked through, then slice into thin strips.
In a small food processor, combine the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and olive oil, pulsing until a smooth, bright green pesto forms.
In the same skillet used for the chicken, add the halved cherry tomatoes and cook for 2 minutes until they begin to soften and blister.
Add the cooked gnocchi and sliced chicken back to the skillet, tossing with the prepared arugula pesto until everything is evenly coated and warmed through.
Serve immediately in a shallow bowl, garnished with the remaining black pepper.