In a small food processor, combine the arugula, walnuts, garlic, parmesan cheese, lemon juice, and half of the olive oil, pulsing until a coarse pesto forms.
Season the chicken breast with sea salt and black pepper.
Heat the remaining olive oil in a large skillet over medium-high heat and sear the chicken until golden and cooked through, then remove to a plate to rest.
In the same skillet, add the cherry tomatoes and cook for 3-4 minutes until they begin to blister and release their juices.
While the tomatoes cook, boil the potato gnocchi in a pot of salted water until they float to the surface, then drain immediately.
Slice the rested chicken into bite-sized pieces.
Add the gnocchi, sliced chicken, and arugula pesto to the skillet with the tomatoes, tossing gently over low heat until everything is well coated and warmed through.