Preheat the oven to 350°F. Spray an 8x8 baking dish or pan with cooking spray, and set it aside.
In a large bowl, whisk together the flour, sugar, light brown sugar, cinnamon, ginger, baking soda, salt, and nutmeg.
Pour in the water, vinegar, vanilla extract, and oil. Use a silicone spatula or wooden spoon to mix everything until the batter is thick and wet and there are no hidden pockets of flour.
Fold in the grated carrots.
Pour the batter into the greased baking dish, place the cake on the middle rack of the oven, and bake for 35-40 minutes or until the top of the cake is brown and the edges of the cake are starting to pull away from the edges of the pan. You can also insert a toothpick in the middle comes out clean or with a few moist crumbs.
After the cake has cooled completely, make the cream cheese frosting. Add the cream cheese, butter, and vanilla extract to a large bowl. Use an electric mixer on medium speed to blend them until smooth. Gradually add the powdered sugar, 1 cup at a time, and once you've added the second cup, increase the mixer to high speed to create a fluffy frosting.
Once the frosting is smooth and creamy, spoon it on top of the cooled cake and spread it out evenly.
Add pecan halves on top, if desired, and serve at room temperature or chilled.
Store the cake in the fridge, covered with plastic wrap or in an airtight container.