Bring a medium pot of salted water to a boil and cook the gnocchi according to package instructions until they float, then drain and set aside.
In a small food processor, combine the arugula, walnuts, parmesan cheese, lemon juice, garlic, and half of the olive oil, pulsing until a coarse pesto forms.
Season the chicken breast with half of the salt and pepper, then slice into bite-sized pieces.
Heat the remaining olive oil in a large skillet over medium-high heat and sear the chicken until golden brown and cooked through, about 5-6 minutes.
Add the cherry tomatoes to the skillet and sauté for 2 minutes until the skins begin to blister and soften.
Reduce heat to low, add the cooked gnocchi and the arugula pesto to the skillet, tossing gently to coat everything evenly.
Season with the remaining salt and pepper, then serve immediately while warm.