Bring a large pot of salted water to a boil and cook the potato gnocchi until they float to the surface, then drain and set aside.
Season the chicken breast with half of the sea salt and black pepper, then pan-sear in a skillet over medium heat until golden and cooked through.
In a food processor, combine the baby arugula, garlic, lemon juice, parmesan cheese, and olive oil, pulsing until a smooth green pesto forms.
Remove the chicken from the skillet to rest, then add the cherry tomatoes to the same pan, cooking until the skins begin to blister and soften.
Slice the rested chicken into thin strips and return them to the skillet along with the cooked gnocchi and the fresh arugula pesto.
Gently toss all ingredients together over low heat until the gnocchi is well-coated and the flavors are melded, then season with the remaining salt and pepper.