Season the chicken breast with half of the sea salt and black pepper.
Heat a non-stick skillet over medium-high heat and sear the chicken until cooked through, then set aside to rest before slicing into bite-sized pieces.
In a food processor, pulse the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and remaining salt and pepper until a coarse paste forms.
Slowly stream in the extra virgin olive oil while pulsing the food processor until the pesto reaches a smooth, emulsified consistency.
Bring a pot of salted water to a boil and cook the potato gnocchi until they float to the surface, which usually takes about 2-3 minutes, then drain.
In the same skillet used for the chicken, lightly sauté the cherry tomatoes over medium heat until the skins begin to blister and soften.
Add the cooked gnocchi and sliced chicken back to the skillet, toss thoroughly with the prepared arugula pesto, and serve immediately.