3 3⁄4 cups all-purpose flour, plus more for pan
1 3⁄4 cups plus 3 tbsp. whole milk
1 3⁄4 cups sugar
1 tbsp. baking powder
2 tsp. vanilla extract
1 1⁄2 cups sweetened shredded coconut
1 cup packed dark brown sugar
Heat the oven to 350° and butter and flour a 3-inch deep, 9-inch springform pan. In a small saucepan, combine 1 3⁄4 cups of the milk with 6 tablespoons of the butter and heat over medium until the butter melts, and then remove from the heat and let cool.
In a large bowl, using a hand mixer, beat the sugar with the eggs until pale yellow and fluffy, about 8 minutes. Stir in the flour, baking powder, and vanilla until smooth and then fold in the milk. Scrape the batter into the prepared pan and place on a parchment paper-lined baking sheet. Bake until golden brown and a toothpick inserted in the middle comes out clean, about 60 minutes. Remove the cake from the oven and increase the oven temperature to 400°.
While the cake is baking, in a 2-quart saucepan, stir the remaining 9 tablespoons butter and 3 tablespoons milk with the coconut and brown sugar and heat over medium until the sugar dissolves and the topping is warm, 4 to 5 minutes.
Spread the topping evenly over the hot cake, return it to the oven, and bake until the topping is bubbling and the coconut begins to toast on top, about 10 minutes. Transfer the cake to a rack and let cool to room temperature. Remove the side of the springform pan, and transfer the cake to a stand or serving platter to serve.