In a small food processor, combine the baby arugula, walnuts, garlic, lemon juice, and 0.25 tbsp of olive oil.
Pulse until a coarse paste forms, adding 1 tbsp of water if needed to reach a smooth consistency.
Season the chicken breast with half of the sea salt and black pepper.
Heat the remaining 0.25 tbsp of olive oil in a non-stick skillet over medium-high heat.
Add the chicken and cook for 6-7 minutes per side until the internal temperature reaches 165°F.
Remove the chicken from the pan and slice into thin strips.
Bring a medium pot of salted water to a boil and cook the potato gnocchi until they float to the surface, about 2-3 minutes.
Drain the gnocchi well and transfer to a large mixing bowl.
Add the sliced chicken and the prepared arugula pesto to the bowl with the gnocchi.
Toss gently until all ingredients are evenly coated and serve immediately.