1. Place tofu on several layers of paper towels. Cover tofu with several more layers of paper towels; top with a cast-iron skillet or other heavy pan. Let stand 30 minutes. Discard paper towels. Cut tofu into 1-inch cubes.
2. Bring 1 1/2 cups water to a boil in a small saucepan; remove from heat. Stir in mushrooms; let stand 30 minutes. Stir in broth and next 6 ingredients (through garlic); bring to a boil. Reduce heat, and simmer 10 minutes, stirring occasionally. Combine remaining 1/4 cup water and cornstarch in a small bowl, stirring with a whisk until smooth. Stir cornstarch mixture, tofu, and bamboo shoots into broth mixture; bring to a boil. Cook 1 minute, stirring occasionally; remove from heat. Slowly drizzle egg into broth mixture (do not stir). Place pan over low heat; cook 1 minute. Stir soup gently to combine. Sprinkle with green onions.