Bring a large pot of salted water to a boil and cook the potato gnocchi according to package directions until they float to the surface, then drain and set aside.
While the water is boiling, dice the chicken breast into bite-sized cubes and season them with half of the sea salt and black pepper.
Heat a large non-stick skillet over medium-high heat and cook the chicken until golden brown and thoroughly cooked, approximately 6 to 8 minutes.
Prepare the pesto by combining the arugula, grated parmesan cheese, olive oil, pine nuts, garlic clove, and lemon juice in a small food processor, pulsing until smooth.
Slice the cherry tomatoes in half and add them to the skillet with the chicken for the final 2 minutes of cooking until they are slightly softened and blistered.
Reduce the skillet heat to low and add the cooked gnocchi and the prepared arugula pesto to the pan.
Gently toss all ingredients together until the gnocchi and chicken are evenly coated in the green sauce, seasoning with the remaining salt and pepper before serving.