In a food processor, pulse the arugula, walnuts, parmesan, garlic, lemon juice, and half of the olive oil until a coarse pesto forms.
Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium-high heat with the remaining olive oil and sear the chicken for 5-7 minutes per side until fully cooked.
While chicken cooks, boil a pot of water and cook the gnocchi for 2-3 minutes or until they float to the surface, then drain.
Remove the chicken from the skillet to rest, then add the cherry tomatoes to the same pan, sautéing until the skins begin to blister.
Slice the chicken and return it to the skillet along with the gnocchi, the prepared pesto, and the remaining salt and pepper.
Toss everything together over low heat for 1 minute until well-coated and fragrant before serving.