Season the chicken breast with half of the salt and pepper, then pan-sear in a non-stick skillet over medium-high heat until cooked through, about 6 minutes per side. Set aside to rest, then slice into strips.
In a small food processor, combine the baby arugula, walnuts, garlic, lemon juice, parmesan cheese, and olive oil. Pulse until a smooth pesto forms.
Bring a pot of salted water to a boil and cook the potato gnocchi according to package directions, usually until they float to the surface. Drain and reserve 2 tablespoons of the cooking water.
In the same skillet used for the chicken, add the cherry tomatoes and cook over medium heat for 3 minutes until the skins begin to blister and soften.
Add the cooked gnocchi and sliced chicken back into the skillet with the tomatoes. Pour in the arugula pesto and the reserved pasta water.
Toss everything together gently over low heat until the sauce coats the gnocchi evenly. Season with the remaining salt and pepper before serving.