Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium heat and cook the chicken until golden and cooked through, about 6-7 minutes per side, then remove and slice.
In a small food processor, combine the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and olive oil, then pulse until a smooth pesto forms.
Bring a pot of salted water to a boil and cook the potato gnocchi according to package instructions until they float, then drain.
In the same skillet used for the chicken, add the cherry tomatoes and cook for 2 minutes until they begin to soften and blister.
Add the cooked gnocchi, sliced chicken, and arugula pesto to the skillet.
Toss everything together over low heat until well coated and heated through.
Season with the remaining sea salt and black pepper before serving hot.