Preheat your oven to 350F and line an 8" square or springform round pan with parchment.
In a large mixing bowl, whisk together the flours, pistachio nuts, arrowroot, baking powder and soda, salt, and sugar. Separately, whisk together the oil, lemon juice, syrup, water, and vanilla. Add the wet ingredients to dry, and whisk together till the batter is smooth. Immediately pour the batter into the baking dish and transfer to the oven. Bake for 45-50 minutes, or until golden brown.
When the cake is ready, allow it to cool for a few hours on a wire rack. When it's room temperature, whisk together the lemon juice and confectioners' sugar. Pour the glaze over the cake and spread it gently with a spatula, if needed. Top with chopped whole pistachio nuts. Transfer the cake to the fridge and allow it to set for another 2-3 hours (or overnight). Cut and serve!
*If your batter is very thick upon mixing, add a little additional water by the tablespoon until it is thick, but can be poured into the baking pan with the help of a spatula and some encouragement.