In a food processor or blender, combine the basil, walnuts, garlic cloves, olive oil, lemon, parmesan, salt and pepper. Pulse until you get a creamy consistency. Reserve.
Boil the peach palm spaghetti for 3 minutes, drain and transfer to a deep serving dish.
Add spoonfuls of the pesto to the spaghetti, mix well and serve immediately (garnished with lemon zest if desired)