In a food processor, pulse the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and half of the sea salt and black pepper until a coarse pesto forms.
Cut the chicken breast into bite-sized pieces and season with the remaining sea salt and black pepper.
Heat the olive oil in a large skillet over medium-high heat and sauté the chicken until golden brown and cooked through.
Bring a pot of salted water to a boil and cook the potato gnocchi until they float to the surface, then drain immediately.
Add the cherry tomatoes to the skillet with the chicken and cook for 2 minutes until the skins begin to blister and soften.
Reduce the heat to low, then add the cooked gnocchi and the prepared arugula pesto to the skillet.
Toss all ingredients gently until the gnocchi is well-coated and the pesto is warmed through before serving.