Pulse the arugula, walnuts, olive oil, parmesan, garlic, and lemon juice in a small food processor until a coarse, vibrant pesto forms.
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, then slice into strips.
While the chicken is cooking, boil the gnocchi in a pot of salted water until they float to the surface, then drain well.
Add the cherry tomatoes to the same skillet used for the chicken and sauté for 2-3 minutes until the skins are blistered and slightly softened.
Reduce the heat to low and return the gnocchi and sliced chicken to the skillet.
Gently fold in the prepared arugula pesto and the remaining salt and pepper, tossing until everything is evenly coated and heated through.