Place boiling water and dried mushrooms in a small bowl. Cover and let stand 30 minutes. Drain mushrooms, reserving liquid.
Heat 1 tablespoon oil in a large saucepan over medium heat. Add shallots to pan; sauté 2 minutes or until tender. Add fresh and rehydrated mushrooms to pan; cook 5 minutes or until tender. Add rice to pan; cook 2 minutes, stirring constantly. Stir in sake; cook 3 minutes or until liquid is nearly absorbed, stirring constantly. Stir in 1/2 cup reserved mushroom liquid and soy sauce; cook 3 minutes or until liquid is nearly absorbed, stirring constantly. Add remaining mushroom liquid, 1/2 cup at a time, stirring constantly until each portion of liquid is absorbed before adding the next. Add water, 1/4 cup at a time, stirring constantly until each portion of water is absorbed before adding the next (about 16 minutes total). Stir in green onions.
Form 9 cakes using a 1/3 cup dry measuring cup coated with cooking spray. Place cakes on a platter. Cover and chill 3 hours. Remove from refrigerator. Place panko in a shallow bowl. Place sesame seeds in a shallow bowl. Dredge each cake in panko on one side and sesame seeds on the other side.
Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add 5 cakes to pan; cook 3 minutes on each side or until lightly browned. Transfer to a platter. Repeat procedure with remaining 1 tablespoon oil and cakes.