Bring a large pot of salted water to a boil.
Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium-high heat with the olive oil and sear the chicken until golden and cooked through, then set aside to rest for five minutes.
In a food processor, pulse 1.5 cups of the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and the remaining salt and pepper until a coarse pesto paste forms.
Cook the gnocchi in the boiling water until they float to the surface, which takes about 2 to 3 minutes, then drain and reserve 2 tablespoons of the cooking water.
Dice the rested chicken into bite-sized pieces.
In the same skillet used for the chicken, combine the gnocchi, diced chicken, arugula pesto, reserved cooking water, and cherry tomatoes.
Toss gently over low heat for 1 minute until the tomatoes just begin to soften and the sauce is creamy, then fold in the remaining 0.5 cup of fresh arugula before serving.