Preheat oven to 350°.
Pour coconut out onto a baking sheet and toast coconut until it's golden. Be VERY careful not to burn it--I recommend checking it after 3 minutes, then again after 5 minutes and so on. You want it to smell and taste like toasted coconut, not burnt! Once coconut is toasted, set it aside.
Combine flour, sugar, baking soda, baking powder, salt, cinnamon, and nutmeg in a bowl and set aside.
Next, melt coconut oil in microwave or over stovetop then set aside, allowing a few minutes for it to come to room temperature.
In a separate bowl, mix coconut milk, syrup, applesauce, vinegar, and vanilla together; then, add the coconut oil.
Combine wet and dry ingredients in the same bowl, then stir the shredded carrots and toasted coconut in by hand.
Line a muffin tin with muffin liners and distribute the batter into the liners. Bake muffins for 20-30 minutes, or until a toothpick comes out clean.
Remove muffins from the oven and while muffins are cooling, make the glaze by whisking milk and powdered sugar together.
Drizzle glaze over top of muffins, sprinkle them with extra shredded coconut and enjoy!