Season the chicken breast with sea salt and black pepper.
Pan-sear the chicken in a skillet over medium-high heat until golden brown and cooked through, then set aside to rest.
Bring a pot of salted water to a boil and cook the potato gnocchi until they float to the surface, then drain.
In a food processor, combine the arugula, walnuts, parmesan cheese, garlic, lemon juice, and extra virgin olive oil, pulsing until a smooth pesto forms.
In the same skillet used for the chicken, sauté the cherry tomatoes until they are blistered and juicy.
Slice the chicken and toss it into the skillet with the gnocchi, tomatoes, and arugula pesto, stirring until everything is evenly coated and warm.