Combine the rice and water in a rice cooker set to white rice. In a small bowl, combine the rice vinegar, sugar, salt and mirin and stir until the sugar is dissolved. Transfer the rice to a bowl, sprinkle with the seasoning and stir using a cutting motion with a flat rice spoon or spatula. Transfer the rice to a small baking sheet and spread into a 1/4-inch thick, even layer. Cover and set aside.

Using a 2-inch cookie cutter, cut the salmon into 12 circles. Using the same cutter, cut the cooled rice into 12 circles. Place the rice onto a platter; put a small dollop of eel sauce on top of each piece of rice. Place a slice of cucumber on top of the rice and then top with the smoked salmon.

Cut triangle eyes and mouth shapes out of the nori sheet and arrange on top of the salmon. Serve with soy sauce for dipping.