Heat vegetable oil in a Dutch oven over medium heat. Add 2 cups onion, and cook 5 minutes, stirring occasionally. Add tofu; cook 5 minutes or until golden brown, stirring frequently.
Add 6 cups water, daikon, and next 7 ingredients (daikon through kombu); bring to a boil. Cover, reduce heat, and simmer 35 minutes.
Combine 6 tablespoons water and cornstarch, stirring with a whisk. Stir cornstarch mixture into stew; bring to a boil. Cook for 2 minutes, stirring constantly. Discard kombu. Stir in sesame oil; sprinkle with green onions.