For the cake:
1 cup quick-cooking oats (not old fashioned or instant oats)
¾ cup water, at room temperature
¾ cup unbleached all-purpose flour
½ tsp. baking soda
½ tsp. baking powder
½ tsp. salt
¼ tsp. ground cinnamon
Dash of ground nutmeg
4 tbsp. unsalted butter, at room temperature
½ cup granulated sugar
½ cup packed light brown sugar
1 large egg, at room temperature
½ tsp. vanilla extract
For the icing:
¼ cup packed light brown sugar
3 tbsp. unsalted butter, melted and cooled
3 tbsp. milk
¾ cup sweetened shredded coconut
½ cup pecans, chopped
To make the cake, preheat the oven to 350˚ F. Line an 8 x 8-inch baking dish with foil and spray lightly with cooking spray. In a medium bowl, combine the oats and water. Stir together, then let sit until absorbed, about 5 minutes. In another bowl, combine the flour, baking soda, baking powder, salt, cinnamon, and nutmeg. In the bowl of a mixer, combine the butter and sugars and beat on medium-high speed until light and fluffy, about 2-4 minutes. Scrape down the bowl as needed. Blend in the egg and vanilla until incorporated. With the mixer on low speed, add the flour mixture in two additions and mix just until incorporated. Add in the soaked oats and mix until combined, about 15 seconds.
Spread the batter into the prepared pan and tap lightly against the counter to remove any air bubbles. Smooth with a spatula. Bake until a toothpick inserted in the center comes out clean, about 30-35 minutes. Let the cake cool slightly in the pan, at least 10 minutes.
While the cake cools, place an oven rack about 9 inches from the broiler and heat the broiler. In a bowl, whisk together the brown sugar, melted butter and milk. Stir in the coconut and pecans. Spread the mixture evenly over the warm cake. Return the pan to the oven and broil until the topping is bubbling and golden, about 3-5 minutes. Let cool in the pan at least 1 hour. Use the foil to lift the cake from the pan, then discard. Slice and serve.