In a medium saucepan combine rice and vegetable broth. Bring to a boil, then reduce heat to simmer until rice is very soft and porridge-like, about 30 minutes. Taste and add salt if needed.
While rice cooks in a medium skillet heat sesame oil over medium. Add mushrooms and cook, stirring, until softened, about 5 minutes. Stir in soy sauce and remove from the heat.
Place eggs in a steamer basket in a large saucepan with ½” water in the bottom. Bring to a boil, cover and steam eggs for 6 minutes. Transfer immediately to an ice water bath to stop cooking. When cool, crack eggs all over on a countertop and peel.