Position racks in the upper and lower thirds of the oven and preheat to 375 degrees . On a baking sheet, toast 1/2 cup of the shredded coconut, 5 to 7 minutes.
In a medium bowl, whisk the flour, baking powder and 3/4 tsp. salt. In a large bowl, using an electric mixer, beat 1 stick of butter with the sugar on high speed until fluffy, about 5 minutes. On low speed, beat in the egg, egg white and 1 tsp. of the vanilla. Beat in the flour mixture and 3/4 cup of the coconut milk in two additions, alternating between the two. Fold in the toasted coconut.
On 2 parchment-lined baking sheets, drop 2-tbsp. scoops of the batter spaced 2 inches apart. Bake, switching pans halfway through, until the cakes' edges are golden, 16 to 18 minutes. Transfer the cakes to a rack. Repeat with remaining batter.
Using an electric mixer, beat 3/4 stick butter, 2 cups confectioners' sugar, the cocoa powder, 1/4 cup coconut milk, 1 tsp. vanilla and 1/4 tsp. salt on low speed to combine. Beat the frosting on high until fluffy, 2 minutes.
Whisk 1 1/2 cups confectioners' sugar into 1/4 cup coconut milk. Working with half the cakes, dunk the rounded side of each into the coconut glaze; sprinkle with shredded coconut. Frost the flat sides of the plain cakes. Top with the glazed cakes.