Wash and peel the carrot. Dice the carrot and mushrooms very finely.
Heat the oil in a large skillet over medium heat. Once hot, add the diced carrot, mushrooms, green peas, and chopped green onions. Cook for about 8 minutes, or until carrots are cooked. Add the soy sauce and cook for one more minute. Remove from heat and set aside.
In a large pot, combine the red rice with the water and salt. Bring to a boil, then reduce heat and let simmer covered for 35-35 minutes, or until the rice is very soft, almost mushy. Season with ground black pepper and sugar. Add the stir-fry vegetables and cook for another 5 minutes.
Serve warm with a sprinkle of ground black pepper and toasted sesame seeds.
This soup will keep for up to two days in the refrigerator. You may add to add more water if reheating to get a soup consistency again.