Season the chicken breast with half of the sea salt and black pepper, then pan-sear in a non-stick skillet over medium heat until cooked through and golden.
Bring a large pot of salted water to a boil and cook the potato gnocchi according to package instructions until they float to the surface.
While the gnocchi cooks, combine the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and extra virgin olive oil in a small food processor, pulsing until a coarse pesto forms.
Halve the cherry tomatoes and lightly sauté them in the same skillet used for the chicken until they just begin to soften and blister.
Slice the cooked chicken breast into thin strips or bite-sized pieces.
Drain the gnocchi and return it to the pot, tossing immediately with the prepared arugula pesto, sliced chicken, and blistered tomatoes.
Season the final dish with the remaining sea salt and black pepper before serving warm.