Add all ingredients to a medium saucepan over medium heat. Cook until the sugar is dissolved, stirring occasionally. Let the syrup cool until it reaches room temperature or is just slightly warm.
Preheat the oven to 325F. Butter and flour the inside of a 9 by 13-inch cake pan.
Whisk together the flour, baking soda, cinnamon, salt, and nutmeg in a large bowl. In a separate large bowl, whisk together the sugar, eggs, oil, buttermilk, vanilla, and crushed pineapple (with juices). Whisk in the shredded carrot. Add the dry ingredients to the wet and stir just to combine, being careful not to over-mix. Stir in the coconut and pecans.
Pour the batter into the prepared pan and spread it out evenly. Bake until a toothpick inserted in the center comes out clean, about 40 to 50 minutes.
While the cake is still hot, poke holes all over the top with a fork or wooden skewer. Pour the cooled cake syrup over the hot cake. Let the cake cool to room temperature, and then refrigerate to chill, about 2 hours.
When the cake is almost fully cooled, make the frosting. Add all ingredients to a large bowl. Use a handheld electric beater to mix until light and fluffy.
Once the cake is cooled, spread the frosting on top.
Before serving, refrigerate for 30 minutes so the frosting can set.
