Make the cake

Butter and lightly flour a 10-inch round cake pan. Set aside.

In a medium bowl, stir together the oats, buttermilk, and milk. Let stand for 20 to 30 minutes. In another bowl, combine the flour, baking powder, baking soda, salt, and nutmeg, and whisk to combine. Set aside.

Position a rack in the center of the oven, and preheat the oven to 350°F. In a large bowl, beat the remaining butter with an electric mixer on medium speed; gradually add the sugar. Beat in the eggs, one at a time. Beat in the maple syrup and vanilla.

Add half of the dry ingredients to the butter mixture, and stir just until combined. Stir in the soaked oat mixture until well combined. Fold in the remaining dry ingredients until well combined.

Pour the batter into the pan, and spread evenly. Bake on the middle oven rack until the cake is golden brown and a toothpick inserted in the center comes out clean, about 40 minutes. Transfer to a wire rack and let cool for 30 minutes. Use a toothpick to poke 20 to 30 holes in the cake.

Position a rack 6 inches from the broiling element, and preheat the broiler.

Make the broiled icing

In a small saucepan, combine the brown sugar, maple syrup, cream, and butter. Bring to a boil, stirring occasionally. Boil for 15 seconds, then remove from the heat and stir in the coconut.

Place the cake on a baking sheet, and evenly spoon the icing over the cake. Broil the cake, until the top is golden brown, 2 to 3 minutes. (Watch carefully and, if necessary, use the baking sheet to adjust the pan to broil evenly.) Transfer the cake to a wire rack, and let cool 10 minutes before serving.

Cover and refrigerate leftovers; serve at room temperature, or briefly warm in a low oven.