You can saute the carrots in little ghee before adding it to the milk.
You can cook on stove top, by boiling milk with the rice till the rice becomes soft, then add the carrots and cook till done. Do not forget to stir occasionally to avoid sticking to the bottom!
Usually, khoya (milk solids) is added for additional richness. If available, add around 100 gm at the end of making the dessert and mix well.
You can add condensed milk instead of sugar.