Preheat the oven to 350°F. Lightly grease a 9×13-inch glass or ceramic baking dish.
Melt the butter. Set aside 6 tablespoons for the cake batter and 2 tablespoons for the topping. Allow it to cool slightly.
Add the eggs to a large mixing bowl and whisk until completely combined.
Whisk in the sweetened condensed milk, evaporated milk, whole milk, vanilla, and 6 tablespoons of melted butter.
In a separate bowl, whisk together the flour, baking powder, and salt.
Gradually add the dry ingredients to the milk mixture, whisking just until combined.
Pour the batter into the prepared baking dish. Bake for 35 to 40 minutes, until the edges have set and the center is still slightly soft.
Mix the shredded coconut, brown sugar, and reserved 2 tablespoons of melted butter in a small bowl.
Carefully remove the cake from the oven. Sprinkle the coconut mixture evenly over the top.
Return the milk cake to the oven and bake for another 18 to 25 minutes, until the topping is deep golden brown and the center is set with a slight jiggle.
Let the cake cool for at least 30 minutes before serving. Serve slightly warm for a soft, creamy texture, or refrigerate for two to three hours for firmer slices.