If you don’t have a sushi mat to use for rolling, wax paper, or a clean kitchen towel works too.
1. To make Sushi Rice: Rinse rice under cold water. Bring rice, 1⅔ cups water, and sake to a boil in saucepan. Reduce heat to low, cover, and simmer 10 minutes. Let rest 10 minutes off heat.
2. Whisk together vinegar, sugar, and salt in bowl. Transfer rice to large bowl, and drizzle half of vinegar mixture over rice, while cutting through it with wooden spoon. Continue adding until all vinegar is absorbed. Cover with damp cloth, and cool.
3. To make Filling: Bring broth, soy sauce, and mirin to a boil in skillet. Add green beans and carrot, cover, and simmer 6 minutes. Transfer to plate with slotted spoon. Add mushrooms to remaining liquid, and bring to a simmer. Cover, and simmer 4 minutes, turning once. Cool, and cut into thin strips.
4. Place 1 nori sheet shiny-side down on sushi mat. Spread one-third Sushi Rice over nori, leaving 1-inch border at top edge. Sprinkle with 1 tsp. sesame seeds. Arrange green beans, carrot, mushrooms, and sprouts in order below center line of rice. Dampen top edge of nori with water; roll up from bottom. Seal with top edge. Repeat with remaining nori, rice, sesame seeds, and vegetables. Cut each roll into 8 slices.