Preheat oven to 350° F and spray a 9x5 loaf pan with nonstick baking spray. Line the bottom of the pan with parchment paper to make it easier to remove.
In a large mixing bowl, add the the milk, canola oil, greek yogurt, eggs, and vanilla and mix together well.
Then add in the brown sugar and mix in thoroughly to combine well.
In a separate medium bowl, add the flour, baking soda, ground cinnamon, and nutmeg and whisk together well.
Shred the carrots into a small bowl. Add approximately 2 tablespoons of the flour mixture and toss to coat them well so they will more evenly blend into other ingredients and not stick together.
Add the dry ingredients to the wet ingredients and stir just until combined, careful not to over-mix or the bread may be too dense.
Add the shredded carrots and gently fold them into the batter.
Pour batter into the prepared loaf pan and bake for 55-60 minutes, or until a toothpick inserted into the center comes out clean.
After baking, let the loaf to cool in the pan for 10 minutes. Then carefully remove and place it on a wire rack to cool completely before icing.