Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium-high heat and sear the chicken for 5-6 minutes per side until cooked through, then set aside to rest before slicing into bite-sized pieces.
Bring a small pot of water to a boil and cook the gnocchi until they float to the surface, about 2-3 minutes, then drain.
In a small food processor, combine 1 cup of the arugula, walnuts, parmesan cheese, garlic, lemon juice, and olive oil. Pulse until a coarse pesto forms.
In the same skillet used for the chicken, add the cherry tomatoes and sauté for 2 minutes until the skins begin to blister.
Add the cooked gnocchi, sliced chicken, and the remaining 0.5 cup of fresh arugula to the skillet.
Stir in the arugula pesto and toss everything together until warmed through and the arugula is slightly wilted.
Garnish with the remaining salt and pepper before serving.