Preheat the oven to 400ºF and line the muffin tin with paper liners.
Combine the oats, flour, cinnamon, nutmeg, baking powder and salt in a large bowl. Set aside
Whisk together the oil, brown sugar, applesauce, eggs and vanilla until smooth. Add the wet ingredients to the dry ingredients and mix until half way combined, and lumps of flour are still showing.
Fold in shredded carrots until evenly distributed and no streaks of flour are showing.
Divide the muffing batter between 12 muffin cups.
Bake at 400ºF for 18-22 minutes until risen and a toothpick inserted in the center comes out clean.
While the muffins are baking, mix together the cream cheese, powdered sugar, milk, and vanilla. With a whisk or an electric hand mixer, mix until smooth and creamy. The glaze should be thick, but will slowly run off a spoon.
Once the muffins are cool enough, spoon a heaping 1 teaspoon of cream cheese icing onto the center of the muffin. You can also drizzle the tops if you prefer.
Chill any leftover glazed muffins for up to five days.