Preheat oven to 325F. Grease an 8 1/2 by 4 1/2 inch loaf pan (original recipe called for a 9x5 inch pan) with coconut oil or your favorite vegan fat, then line with a strip of parchment paper and grease again.
Pulse pistachios in a food processor until coarsely ground, then set aside about 1 tablespoon. Pulse until finely ground.
In a mixing bowl, whisk or stir together lemon zest, flour, granulated sugar, baking powder, salt, and remaining pistachios. Make sure all of the dry ingredients are very thoroughly mixed.
In a small bowl, whisk olive oil, 2 Tbsp. lemon juice, 1/3 cup melted coconut oil, and 1/2 cup water.
Pour oil mixture into dry ingredients and fold with a rubber spatula to combine, being very careful not to overmix, as this will result in a gummy texture. The batter is fairly thick.
Transfer to greased pan and spread evenly.
Bake cake until golden brown all over, it springs back when gently pressed, and a tester inserted into the center comes out clean, 50–60 minutes (mine took a full 60 minutes in the 8 1/2 by 4 1/2 inch pan).
Let cool completely for at least two hours.
Using parchment overhang, lift cake out of pan and place on rack. Peel away parchment.
To make the icing put 1 1/2 to 2 cups powdered sugar in a bowl and begin adding lemon juice one tablepoon at a time, stirring after each addition, until the icing is a pourable consistency. Drizzle over and down the sides, then sprinkle with reserved ground pistachios.