In a small food processor, pulse the baby arugula, walnuts, garlic, lemon juice, and olive oil until a smooth, vibrant pesto forms.
Season the chicken breast evenly with half of the sea salt and black pepper.
Heat the ghee in a large skillet over medium-high heat and sear the chicken for 6-7 minutes per side until golden and cooked through.
Remove the chicken from the pan, let it rest for 5 minutes, then slice it into bite-sized pieces.
Bring a large pot of salted water to a boil and cook the gnocchi until they float to the surface, then drain immediately.
In the same skillet used for the chicken, add the cherry tomatoes and sauté for 2 minutes until the skins begin to burst and soften.
Turn off the heat and add the cooked gnocchi, sliced chicken, and arugula pesto to the skillet, tossing gently to coat.
Season with the remaining sea salt and black pepper before serving in a warm bowl.