Preheat oven to 350 degrees F (175 degrees C). Lightly grease 9×13 baking dish.
Place the carrots, eggs, and oil in a blender or bowl of a food processor. Process until carrots are finely chopped. Pour the carrot mixture into a mixing bowl. Stir in 2 cups sugar until well blended. Stir in the flour and baking powder; mix until well blended. Pour the batter into the prepared baking dish.
Bake in preheated oven until top springs back when lightly touched, about 40 minutes.
Meanwhile, make the icing by placing the butter, 1 cup sugar, instant hot chocolate drink mix, and milk in a pan. While stirring, heat to almost boiling over medium-high heat until mixture thickens. When the cake is done, remove from the oven and immediately spread the icing evenly over the top.
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