Season the chicken breast evenly 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 golden and cooked through, then set aside to rest before slicing into strips.
In a small food processor, combine the arugula, walnuts, parmesan cheese, lemon juice, garlic, and olive oil, pulsing until a coarse, vibrant pesto forms.
Bring a small pot of salted water to a boil and cook the gnocchi until they float to the surface, which should take about 2-3 minutes, then drain immediately.
In the same skillet used for the chicken, add the cherry tomatoes and sauté for 3-4 minutes until the skins begin to blister and soften.
Add the cooked gnocchi, sliced chicken, and arugula pesto to the skillet with the tomatoes, tossing gently over low heat to coat every piece before serving.