In a small food processor, pulse the arugula, walnuts, garlic, lemon juice, and parmesan cheese until finely chopped.
Season the chicken breast with half of the sea salt and black pepper, then sear in a non-stick skillet over medium-high heat until cooked through, about 6 minutes per side.
While the chicken cooks, boil the potato gnocchi in salted water according to package directions until they float, then drain.
Remove the chicken from the skillet and set aside to rest; in the same skillet, add the cherry tomatoes and cook until they begin to blister and soften.
Slice the chicken into bite-sized pieces and return it to the skillet along with the cooked gnocchi.
Add the arugula pesto and olive oil to the skillet, tossing everything together over low heat until well-coated and fragrant.
Season with the remaining sea salt and black pepper before serving in a warm bowl.