Prepare the pesto by pulsing the arugula, walnuts, parmesan cheese, 1 teaspoon of olive oil, garlic, and lemon juice in a food processor until a coarse paste forms.
Season the chicken breast with sea salt and black pepper.
Heat the remaining 1 teaspoon of olive oil in a skillet over medium-high heat and sear the chicken until golden and cooked through, about 5-6 minutes per side.
While the chicken cooks, bring a pot of salted water to a boil and cook the gnocchi until they float to the surface, then drain immediately.
Add the cherry tomatoes to the skillet with the chicken during the last 3 minutes of cooking, allowing them to blister and soften.
Slice the cooked chicken into bite-sized pieces and return to the skillet.
Add the drained gnocchi and the arugula pesto to the skillet, tossing everything together over low heat until well combined and heated through.