Bring a large pot of salted water to a boil and cook the potato gnocchi according to package directions until they float to the surface.
While gnocchi cooks, 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 for 5-7 minutes per side until fully cooked, then slice into strips.
In a small food processor, combine the baby arugula, chopped walnuts, garlic, lemon juice, grated parmesan, and olive oil.
Pulse the pesto ingredients until a coarse paste forms, adding a teaspoon of water if needed to reach your desired consistency.
Halve the cherry tomatoes and lightly sauté them in the same skillet used for the chicken for 2 minutes until just softened.
Drain the gnocchi, reserving two tablespoons of the cooking water.
Toss the gnocchi, sliced chicken, and sautéed tomatoes with the arugula pesto and reserved pasta water until everything is evenly coated.
Season with the remaining sea salt and black pepper before serving in a shallow bowl.