To make the cake, in a medium saucepan over medium, heat the butter, occasionally swirling the pan and scraping the bottom with a wooden spoon, until dark amber and the milk solids at the bottom are almost black, 8 to 10 minutes.

Transfer to the bowl of a stand mixer, making sure to scrape in all of the milk solids. Cool until the butter is opaque, spreadable and cool to the touch, about 1 hour.

While the butter cools, heat the oven to 350°F with a rack in the middle position. Spread the shredded coconut in a 9-by-5-inch loaf pan and toast in the oven until golden brown, 5 to 7 minutes, stirring once about halfway through.

Measure 1 tablespoon of the toasted coconut into a small bowl, then transfer the remainder to a medium bowl; set both aside. Let the pan cool.

Mist the loaf pan with cooking spray. Line it with an 8-by-12-inch piece of kitchen parchment, fitting the parchment into the bottom and up the pan's long sides; mist the parchment with cooking spray.

To the medium bowl with the coconut, whisk in both flours, the baking powder and salt. In a liquid measuring cup or small bowl, stir together the buttermilk and vanilla.

Add the white sugar to the cooled browned butter. In the stand mixer with the paddle attachment, mix the butter and sugar on medium until well combined, about 2 minutes, scraping the bowl about halfway through.

With the mixer running on medium, add the eggs one at a time, scraping the bowl after the first 2 additions. Beat on medium until the mixture is shiny and lightened in color, about 1 minute. With the mixer running on low, add half of the flour, then the buttermilk mixture, followed by the remaining flour mixture.

Mix on low for about 10 seconds, then stop the mixer. Using a silicone spatula, fold the batter just until the flour is incorporated, scraping the bottom of the bowl to ensure no pockets of butter or flour remain.

Transfer the batter to the prepared pan and smooth the surface. Bake until the top is deeply browned and a toothpick inserted into the center comes out with a few small crumbs attached, 75 to 80 minutes.

While the cake bakes, make the syrup. In a small microwave-safe bowl, stir together the coconut milk, white sugar and 2 tablespoons water. Microwave on high for 30 seconds, stirring once about halfway through to ensure the sugar is dissolved. Set aside to cool.

When the cake is done, cool in the pan on a wire rack for 15 minutes. Using the parchment overhang as handles, remove the cake from the pan and set on the rack. With a toothpick, poke holes in the top of the cake at 1-inch intervals.

Brush all of the syrup onto the cake, allowing it to soak in. Cool to room temperature, about 2 hours. Remove and discard the parchment.

To make the glaze, in a medium bowl, whisk the powdered sugar, coconut milk and salt until smooth. Spoon over the cooled cake, spreading it to cover the surface and allowing it to drip down the sides slightly.

Sprinkle with the reserved 1 tablespoon toasted coconut. Allow the glaze to dry for at least 5 minutes before serving.