Preheat the to 400 degrees Fahrenheit. Grease a muffin tin or line with paper cups for easy removal.
Using an electric hand mixer, beat the softened cream cheese and sugar until smooth. Set aside.
In a medium-sized mixing bowl, whisk together the flour, sugar, light brown sugar, baking powder, baking soda, cinnamon, ginger, and salt. Set aside.
In another mixing bowl, whisk together eggs, water and oil. Add the egg mixture to the flour mixture and mix until incorporated. Stir in the grated carrots.
Spoon a heaping tablespoon of the carrot cake batter into the bottom of each muffin cup. Add 1 tablespoon of the cream cheese mixture to the center, then add 2 Tablespoons of the carrot cake mixture on top. Repeat until you have about 10 muffins.
Sprinkle the tops with the coarse sugar and bake for 17-20 minutes or until a toothpick inserted in the middle comes out clean and the tops are firm. Cool the muffins for 20 minutes before serving.