Bring a medium pot of salted water to a boil for the gnocchi.
In a small food processor or blender, combine 1.5 cups of the baby arugula, walnuts, garlic, lemon juice, grated parmesan, and 1 teaspoon of olive oil; pulse until a coarse pesto forms.
Dice the chicken breast into bite-sized pieces and season with half of the sea salt and black pepper.
Heat the remaining 1 teaspoon of olive oil in a large skillet over medium-high heat and sear the chicken until golden and cooked through, about 5-6 minutes.
Add the halved cherry tomatoes to the skillet with the chicken and cook for 2-3 minutes until they begin to soften and blister.
Drop the gnocchi into the boiling water and cook until they float to the surface, which usually takes about 2 minutes, then drain immediately.
Add the cooked gnocchi and the prepared arugula pesto to the skillet with the chicken and tomatoes.
Toss everything together over low heat for 1 minute to coat, stirring in the remaining 0.5 cup of fresh arugula and the rest of the salt and pepper before serving.