1. Heat oven to 350 degrees F. Coat a large, rimmed baking sheet with butter. Cover the bottom with parchment paper.
2. In a small bowl, whisk together cake flour, baking powder, and salt. Set aside. In a large bowl, cream the butter and sugar. Add eggs, one at a time, blending after each addition. Blend in the vanilla and coconut extracts.
3. Add one third of the coconut milk to the batter and blend. Add one third of the flour mixture and blend well. Continue alternating between the milk and the flour mixture until all ingredients have been incorporated into the batter.
4. Pour batter into the prepared baking sheet and spread evenly. Bake until golden brown and a toothpick comes out clean, about 25 minutes. Let cool completely. Leave the oven on.
5. While the cake cools, toast coconut flakes on a baking sheet in the oven until golden, stirring halfway through, about 6 minutes. Transfer to a plate to cool.
6. Beat cream cheese, butter, vanilla, and coconut extract on medium until smooth. Gradually add confectioners' sugar and blend until creamy.
7. Use a 1 1/2-inch round cookie cutter to cut 32 rounds. To assemble, generously top a cake round with frosting. Add another cake layer, then top with frosting and a sprinkle of toasted coconut.