Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium-high heat and cook the chicken for 6-7 minutes per side until golden and the internal temperature reaches 165°F.
While chicken cooks, bring a pot of salted water to a boil and cook the gnocchi until they float to the surface, then drain immediately.
In a small food processor, combine the arugula, walnuts, olive oil, parmesan cheese, garlic, lemon juice, and the remaining salt and pepper; pulse until a coarse pesto forms.
Remove the chicken from the skillet, let it rest for 3 minutes, then slice into thin strips.
In the same skillet used for the chicken, toss the cherry tomatoes over medium heat for 2 minutes until the skins begin to blister and soften.
In a large mixing bowl, combine the cooked gnocchi, sliced chicken, and blistered tomatoes, then fold in the arugula pesto until everything is evenly coated.