Bring a pot of salted water to a boil and cook the potato gnocchi according to package instructions until they float, then drain and set aside.
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 to 6 minutes per side until golden and cooked through, then slice into thin strips.
In a small food processor, pulse the baby arugula, walnuts, garlic, lemon juice, parmesan cheese, and olive oil until a coarse, bright green paste forms.
In the same skillet used for the chicken, add the cherry tomatoes and sauté for 2 minutes until the skins begin to blister and soften.
Add the cooked gnocchi and sliced chicken back into the skillet with the tomatoes and toss to combine.
Remove the skillet from the heat and fold in the arugula pesto, stirring gently to coat everything evenly, and season with the remaining salt and pepper.