Bring a large pot of salted water to a boil.
Season the chicken breast with half of the sea salt and black pepper.
Heat a skillet over medium-high heat with 0.25 tbsp of the olive oil and sear the chicken until golden and cooked through, then slice into strips.
In a small food processor, combine the baby arugula, walnut halves, garlic, lemon juice, and the remaining sea salt and black pepper.
Pulse the mixture while drizzling in 0.25 tbsp of the olive oil until a rustic pesto forms.
Cook the potato gnocchi in the boiling water until they float to the surface, then drain.
In the same skillet used for the chicken, add the remaining 0.25 tbsp of olive oil and the cherry tomatoes, sautéing until they begin to blister.
Add the gnocchi, sliced chicken, and arugula pesto to the skillet, tossing gently to coat before serving topped with the parmesan cheese.