FOR THE CAKE:

1. Combine oats and water in medium bowl and let sit until water is absorbed, about 5 minutes.

2. Adjust oven rack to middle position and heat oven to 350 degrees. Cut two 16-inch lengths aluminum foil and fold both lengthwise to 5-inch widths. Spray 8- by 8-inch metal baking dish with nonstick cooking spray. Fit foil pieces into baking dish to form a sling for removing the cake later, one overlapping the other, pushing them into corners and up sides of pan; allow excess to overhang pan edges. Spray foil lightly with nonstick cooking spray.

3. In another medium bowl, whisk flour, baking soda, baking powder, salt, cinnamon, and nutmeg together.

4. In another bowl, beat butter and sugars on medium speed until combined and mixture has consistency of damp sand, scraping down bowl with rubber spatula halfway through mixing. Add egg and vanilla; beat until combined. Add flour mixture in 2 additions and mix until just incorporated. Add soaked oats and mix until combined.

5. Give batter final stir with rubber spatula to make sure thoroughly combined. Transfer batter to prepared pan and lightly tap against counter 3 or 4 times to dislodge any large air bubbles; smooth surface with spatula. Bake cake until toothpick inserted into center comes out with few crumbs attached, 30 to 35 minutes (start checking at 25 minutes), rotating pan halfway through baking. Let cake cool slightly in pan, at least 10 minutes.

6. FOR THE BROILED ICING: While cake cools, adjust oven rack about 9 inches from broiler element and heat broiler. In medium bowl, whisk brown sugar, melted butter, and milk together; stir in coconut and pecans. Spread mixture evenly over warm cake. Broil until topping is bubbling and golden, 3 to 5 minutes.

6. Let cake cool in pan 1 hour. Following illustration 2, transfer cake to serving platter, then discard foil. Cut cake into squares and serve.