In a bowl, mix soy sauce, mirin, sake, sugar, and dashi. Set aside.
Heat skillet over medium heat, add vegetable oil. Cook onions until soft (about 5 minutes).
Add sliced beef to skillet, and cook until browned (3-5 minutes).
Pour soy sauce mixture over beef and onions, stir to combine. Bring to a boil, then simmer for 2-3 minutes, or until the sauce thickens.
In a separate pan, cook eggs sunny-side up or over-easy.
Place cooked rice in a bowl, top with beef and onions.
Optionally, pour the remaining sauce over rice and beef.
Add cooked egg on top and garnish with sliced green onions, if desired.
Your delicious beef ojyu is now ready to be enjoyed!