Season the chicken breast with half of the sea salt and black pepper.
In a skillet over medium-high heat, cook the chicken until golden and the internal temperature reaches 165°F, then slice into bite-sized pieces.
Boil the potato gnocchi in salted water according to package directions until they float, then drain immediately.
In a small food processor, pulse the arugula, walnuts, garlic, lemon juice, and olive oil until a coarse pesto forms.
In the same skillet used for the chicken, add the cherry tomatoes and cook for 2-3 minutes until they begin to blister and soften.
Add the cooked gnocchi, sliced chicken, and arugula pesto to the skillet, tossing gently to coat everything evenly.
Season the dish with the remaining sea salt and black pepper before serving warm.