Step 1
Bring water and rice to a boil in a large pot. Reduce heat to medium-low cover and simmer until the rice is tender and liquid has been absorbed about 25 minutes. Remove pot from heat and keep covered for 10 minutes.
Step 2
Stir 5 1/2 tablespoons rice vinegar sugar and salt together in a microwave-safe bowl; microwave until vinegar mixture is warmed 30 to 45 seconds. Stir well. Add vinegar mixture to rice and toss thoroughly to coat each grain of rice; cool completely.
Step 3
Heat 1 teaspoon vegetable oil in a small skillet over medium heat; cook eggs in hot oil until firm 3 to 5 minutes. Transfer eggs to a plate and cut into strips.
Step 4
Heat 1 tablespoon vegetable oil sake 1 tablespoon soy sauce and sesame oil in a skillet over medium heat; fry eggplant in oil mixture until softened and lightly charred 5 to 10 minutes. Transfer eggplant to a paper towel-lined plate.
Step 5
Mix carrot 1 tablespoon rice vinegar and 1 tablespoon soy sauce together in a microwave-safe bowl; microwave until carrot is softened 1 to 2 minutes. Drain.
Step 6
Bring a large pot of lightly salted water to a boil; add asparagus and cook until bright green 2 to 3 minutes. Drain and immediately immerse asparagus in ice water for several minutes to stop the cooking process. Drain.
Step 7
Cut avocado into 8 slices and drizzle lemon juice over slices in a bowl.
Step 8
Lay nori sheets on a flat work surface wet hands and spread 3/4 to 1 cup rice onto each sheet. Press rice into a thin layer leaving 1/2-inch exposed nori on 1 long side.
Step 9
For California rolls layer crabmeat avocado cucumber and carrot atop one another in a thin strip along the edge opposite the uncovered edge on 3 nori sheets.
Step 9
For eggplant and avocado rolls layer eggplant and avocado atop one another in a thin strip along the edge opposite the uncovered edge on 3 nori sheets.