Preheat your oven to 350 degrees Fahrenheit (175 degrees Celsius). Grease a 9x13-inch round baking pan with butter or non-stick cooking spray.
In a large bowl, combine the sugar, oil, eggs, and baby carrots. Mix until everything is well combined. In a separate bowl, combine the flour, baking soda, salt, and cinnamon.
Slowly add in the dry ingredients to the wet ingredients and mix until everything is well combined. Pour the batter into the round cake pan and spread it evenly with a rubber spatula.
Bake for 45 minutes, or until a toothpick inserted into the center of the cake comes out clean. Let cool completely before adding the cream cheese icing.
To make the cream frosting, combine softened butter, cream, and vanilla in a medium mixing bowl. Blend until completely smooth and combined. If the icing is too thick, add drops of milk to reach desired consistency. Cover with plastic wrap and store at room temperature.
Start by placing the first layer on a serving plate and evenly cover the top of the cake with frosting. Place the second layer on top, face down, and press firmly in place. Spread the rest of the frosting over the top and sides of the cake to evenly coat.