Season the chicken breast with half of the sea salt and black pepper.
Heat avocado oil in a large skillet over medium-high heat and sear the chicken for 5-6 minutes per side until golden and cooked through.
Remove chicken from the pan, let it rest for 3 minutes, then slice into bite-sized pieces.
Bring a pot of salted water to a boil and cook the gnocchi until they float to the surface, then drain and reserve 2 tablespoons of the cooking water.
In a small food processor, pulse the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and extra virgin olive oil until a coarse pesto forms.
In the same skillet used for the chicken, add the cherry tomatoes and sauté for 2-3 minutes until the skins begin to blister and soften.
Add the cooked gnocchi, sliced chicken, and arugula pesto to the skillet, tossing gently with the reserved pasta water to coat everything evenly before serving.