Bring a small 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.
Season the chicken breast with half of the sea salt and black pepper, then pan-sear in a non-stick skillet over medium-high heat until golden brown and cooked through, about 6 minutes per side.
While the chicken cooks, prepare the pesto by placing the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and olive oil into a small food processor.
Pulse the pesto ingredients until the mixture is well combined but still retains a slight texture.
Remove the chicken from the skillet and slice into bite-sized strips.
In the same skillet, add the cherry tomatoes and cook for 2-3 minutes until the skins begin to blister and pop.
Add the cooked gnocchi and sliced chicken back into the skillet with the tomatoes.
Stir in the arugula pesto and the remaining sea salt and black pepper, tossing everything together for 1 minute until the sauce is warmed and evenly coats the gnocchi.
Serve immediately while the gnocchi are light and the pesto is bright green.