Bring a medium pot of salted water to a boil and cook the 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.
Heat a non-stick skillet over medium-high heat and sear the chicken breast for 6-7 minutes per side until the internal temperature reaches 165°F.
While the chicken cooks, place the baby arugula, walnuts, parmesan cheese, garlic, lemon juice, and the remaining salt and pepper into a small food processor.
Pulse the mixture until finely chopped, then slowly drizzle in the olive oil while continuing to pulse until a smooth pesto forms.
Remove the chicken from the skillet and let it rest for 3 minutes before slicing 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 soften.
Add the cooked gnocchi and sliced chicken back into the skillet with the tomatoes.
Turn off the heat and stir in the arugula pesto, tossing everything together until the gnocchi and chicken are evenly coated and vibrant.
Serve immediately in a shallow bowl.