Wash rice in a pan with cold water until water is clear. Drain and place rice in a covered heavy pot or saucepan with 2 1/3 cups water. Soak for 1 hour.
Bring rice and water to a boil. Lower heat and simmer until water is absorbed (about 25 minutes). Turn off heat and let sit for 10 minutes. Remove rice from pot and place in a large serving bowl.
While rice is cooking, mix rice vinegar, sugar, and salt in a small bowl until sugar and salt dissolve. Sprinkle mixture over rice and gently fold it into the rice with a wooden spoon or spatula. Leave to cool.
Soak shiitakes in 1 1/2 cups warm water for 30 minutes. Remove from water and set water aside. Cut off mushroom stems and squeeze mushrooms dry. Cut into thin shreds.
In a small saucepan, combine 1/2 cup of mushroom-soaking liquid, soy sauce,
1 1/2 tablespoons sugar, mushrooms, and carrot sticks. Cover and bring to a boil. Remove lid and simmer for 2 minutes, stirring constantly. Remove from heat, drain, and set aside.
Sprinkle lemon juice over crab meat and let sit for 5 minutes. Squeeze extra liquid out of crab meat, break into small pieces, and set aside.
Boil green peas until they are tender. Remove from heat, drain, and set aside.
Beat eggs with 1/2 teaspoon sugar and pinch of salt. Heat a frying pan lightly coated with vegetable oil and pour in half of egg mixture, tilting pan to make a thin omelet. Fry over low heat for 30 seconds or until surface of omelet is dry. Carefully remove omelet with a spatula and place on a cutting board. Repeat with remaining mixture. When omelets are cool enough to handle, cut into thin strips.
With a wooden spoon or spatula, fold about half of the mushrooms, carrot, peas, and crab meat into the rice. Scatter the remainder, along with omelet strips, on top and serve at room temperature.