Bring mirin, soy sauce, and sake to a boil in a pot over medium heat. Add onion; cook and stir until onion is translucent, about 5 minutes. Stir in beef; simmer until no longer pink and cooked through, about 5 minutes. Spoon beef and onion over rice using a slotted spoon.
Place eggs in the pot; bring to a boil. Cook until soft boiled, 5 to 7 minutes. Peel; add to bowls. Garnish with pickled ginger.