Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium-high heat and cook the chicken for 6-7 minutes per side until golden and cooked through, then set aside to rest and slice into strips.
In a food processor, pulse the arugula, pine nuts, garlic, parmesan cheese, lemon juice, and olive oil until a smooth, vibrant pesto forms.
Boil a pot of salted water and cook the potato gnocchi according to package directions until they float to the surface, then drain immediately.
In the same skillet used for the chicken, add the cherry tomatoes and cook for 2-3 minutes until they begin to blister and soften.
Add the cooked gnocchi and sliced chicken back into the skillet, then stir in the arugula pesto until everything is evenly coated and warmed through.
Season the dish with the remaining sea salt and black pepper before serving in a shallow bowl.