In a food processor, pulse the arugula, walnuts, garlic, lemon juice, and half the sea salt and black pepper until finely chopped.
Slowly drizzle in half the olive oil while pulsing until a thick pesto forms, then stir in the parmesan cheese.
Season the chicken breast with the remaining salt and pepper, then sear in a skillet over medium heat with the remaining olive oil until golden and cooked through.
Remove the chicken from the pan to rest for 3 minutes, then slice into thin strips.
Boil the potato gnocchi in salted water according to package directions until they float, then drain immediately.
In the same skillet used for the chicken, toss the cherry tomatoes for 2 minutes until the skins begin to blister.
Add the cooked gnocchi and sliced chicken back to the skillet, then stir in the arugula pesto until everything is evenly coated and warmed through.