Bring rice and the water to the boil in a large saucepan over medium-high heat. Reduce heat; simmer, covered, for 30 minutes or until rice is just tender. Remove from heat; stand, covered, until cool enough to handle.
Meanwhile, combine mirin, sauce, sugar, vinegar and 2 teaspoons of the oil in a small bowl; set aside.
Place half the rice, half the white part of the green onion and 1 teaspoon of the oil in a food processor; pulse until rice is coarsely chopped and sticky. Transfer rice mixture to a large bowl. Repeat with remaining rice, white part of the green onion and another 1 teaspoon of oil. Stir rice mixture to combine. Shape rice mixture into 16 patties. Brush rice cakes with 1 tablespoon of the oil.
Heat a large frying pan over medium heat; cook rice cakes, in batches, for 5 minutes each side or until golden and crisp.
Meanwhile, heat remaining oil in a wok or large frying pan over medium-high heat; cook mushrooms until almost tender. Add zucchini and peas; cook for 1 minute or until tender. Add mirin mixture; toss to combine.
Serve rice cakes with stir-fried vegetables, sprinkled with seaweed and remaining green onion.