Preheat the oven to 350 degrees F (175 degrees C). Grease an 8 1/2 by 4 1/2 by 2 1/2-inch loaf pan, line the bottom with parchment paper, then grease the paper and flour the pan.
In the bowl of an electric mixer fitted with the paddle attachment, combine the Unsalted Butter (1/2 cup) and Granulated Sugar (1 cup). Beat on medium speed for 3-5 minutes, until light and fluffy.
With the mixer on medium speed, add the Large Egg (3) one at a time, scraping down the bowl after each addition. Beat for one more minute, until the batter is pale and thick.
In a medium bowl, sift together the All-Purpose Flour (1 1/4 cup), Baking Powder (1/2 teaspoon), Kosher Salt (1/2 teaspoon), and Baking Soda (1/4 teaspoon). Add the Unsweetened Shredded Coconut (1/4 cup) and set aside.
Place the Shelled Unsalted Pistachios (1/3 cup) in the bowl of a food processor and process until coarsely chopped. Remove 1 1/2 tablespoons and set aside for garnish. Continue processing until the pistachios are very finely ground. Measure to make sure you have 1/4 cup, and add it to the bowl with the dry ingredients.
In another bowl, combine Full-Fat Coconut Milk (1/2 cup), zest of the Lemon (1 tablespoon), Lemon Juice (3 tablespoon), and Vanilla Extract (1/2 teaspoon). Add the wet and dry mixtures alternately to the batter, beginning and ending with the dry. Do not overmix.
Transfer to the prepared pan and bake for 45-50 minutes, until a cake tester comes out clean.
Allow to cool in the pan for 10 minutes, then invert onto a rack set over a tray to continue cooling.
In a small saucepan, combine the Granulated Sugar (1/4 cup), Full-Fat Coconut Milk (2 tablespoon), and Lemon Juice (2 tablespoon). Bring to a simmer, stirring to dissolve the sugar, then remove from the heat.
While the cake is still warm, slowly pour the syrup over the top of the cake. Sprinkle with the reserved pistachios and Unsweetened Shredded Coconut (1 tablespoon), cool completely, and serve.