Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium-high heat and sear the chicken until cooked through, about 6-7 minutes per side, then set aside to rest before slicing.
Bring a small pot of salted water to a boil and cook the gnocchi according to package directions until they float, then drain.
In a small food processor, combine the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and olive oil, pulsing until a smooth pesto forms.
In the same skillet used for the chicken, add the cherry tomatoes and cook for 2-3 minutes until the skins begin to blister and burst.
Add the cooked gnocchi and sliced chicken back into the skillet with the tomatoes.
Remove from heat and stir in the arugula pesto, tossing well to coat all ingredients evenly before serving.