Bring a medium pot of salted water to a boil and cook the potato gnocchi according to package instructions until they float to the surface; drain and set aside.
Season the chicken breast with half of the sea salt and black pepper, then sear in a non-stick skillet over medium-high heat for 6-7 minutes per side until fully cooked.
While the chicken is cooking, prepare the pesto by pulsing the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and olive oil in a food processor until a coarse paste forms.
Remove the chicken from the skillet, let it rest for 3 minutes, and then slice it into bite-sized strips.
In the same skillet used for the chicken, add the cherry tomatoes and cook for 2-3 minutes over medium heat until they begin to soften and the skins start to blister.
Reduce the heat to low, then add the cooked gnocchi, sliced chicken, and arugula pesto to the skillet, tossing gently for 1 minute until the sauce is warmed and everything is evenly coated.