Season the chicken breast with half of the sea salt and black pepper on both sides.
In a large skillet over medium-high heat, cook the chicken until golden and the internal temperature reaches 165 degrees Fahrenheit, then set aside to rest for five minutes before slicing.
Bring a pot of salted water to a boil and cook the potato gnocchi until they float to the surface, which usually takes about 2 to 3 minutes.
While the gnocchi cooks, combine the arugula, pine nuts, garlic, lemon juice, parmesan cheese, and the remaining salt and pepper in a food processor, pulsing while drizzling in the olive oil until a rustic pesto forms.
Drain the gnocchi and add them to the same skillet used for the chicken, searing them for 2 minutes until the edges are slightly crisp and golden brown.
Slice the rested chicken and toss it into the skillet with the gnocchi, the arugula pesto, and the halved cherry tomatoes, stirring gently over low heat until everything is evenly coated and warmed through.