Bring a 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 evenly with half of the sea salt and black pepper.
Heat the avocado oil in a large skillet over medium-high heat and sear the chicken for 6 to 7 minutes per side until golden brown and cooked through.
While the chicken is searing, place the baby arugula, walnuts, garlic, lemon juice, parmesan cheese, and extra virgin olive oil into a food processor.
Pulse the food processor until the ingredients form a coarse, vibrant green pesto paste.
Remove the chicken from the skillet and slice it into thin, bite-sized strips.
In a large mixing bowl, combine the warm gnocchi, sliced chicken, and the fresh arugula pesto, tossing thoroughly to coat.
Finish with the remaining sea salt and black pepper before serving immediately.