Make the cake: Heat the oven to 350 degrees and move the oven rack to the middle position. Grease an 8 inch square baking pan. Whisk the flour, baking soda, salt, cinnamon and nutmeg together in a medium bowl. Combine the sugars and butter together in the bowl of a stand mixer. Mix on medium speed using the paddle attachment until light and fluffy, about 3 minutes. Add the egg and vanilla extract and continue mixing until thoroughly combined.
Reduce the speed to low and add one-third of the flour mixture followed by one-half of the milk. Repeat with another third of the flour mixture and the remaining half of the milk mixture. Mix in the last of the flour mixture until just incorporated. Gently mix in the oats.
Scrape the batter into the prepared cake pan. Smooth the top with an offset spatula. Bake for 35-45 minutes or until the cake is golden brown and a tester inserted in the center of the cake comes out clean.
About 10 minutes before the cake is done, make the icing: Melt the butter in a nonstick skillet over medium heat. Add the brown sugar and salt and cook, stirring constantly until bubbling. Add the cream and coconut and bring to a simmer. Remove from the heat and set aside.
Once the cake comes out of the oven, immediately spread the coconut topping evenly over the top of the warm cake. Set the oven to broil, leaving the oven rack in the middle position. Return the cake to the oven and broil until the coconut mixture is browned and bubbly, rotating the pan as needed, about 2 minutes. Keep a close eye on the cake while it is broiling to prevent it from over browning (I leave the oven door propped open slightly and watch it while it broils). Cool to room temperature and serve.