In a medium saucepan bring the water to a boil over high heat. Stir in the carrot and mushrooms and boil for about 7 minutes. Add the noodles and the flavouring packet; stir to break up the noodles. Reduce heat to medium and simmer for 3 minutes. Slowly pour in the egg and stir for 30 seconds until the egg has cooked. Stir in the milk.