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 golden and cooked through, then slice into thin strips.
Bring a large pot of salted water to a boil and cook the potato gnocchi until they float to the surface, then drain immediately.
In a food processor, combine the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and the remaining salt and pepper.
Pulse the pesto ingredients while slowly drizzling in the olive oil until a coarse, vibrant paste forms.
In the same skillet used for the chicken, lightly sauté the halved cherry tomatoes for 2 minutes until they begin to soften.
Combine the cooked gnocchi, sliced chicken, and sautéed tomatoes in a large bowl and toss thoroughly with the fresh arugula pesto.