Cut the beef slices into strips less than half an inch in width.
Peel and thinly slice the onion.
Heat two tablespoonfuls of cooking oil in a frying pan and cook the eggs, sunny side up. You can do this one by one, or cook the eggs at the same time. It all depends on how large your frying pan is. When the eggs are done, set aside and keep warm.
Heat the remaining two tablespoonfuls of cooking oil in the pan. Cook the onion slices over medium-low heat until softened, about three minutes.
Turn up the heat to high. Add the beef to the onion slices and stir to break up any clumps. Cook just until the meat is no longer pink.
Pour in the soy sauce, mirin, sake and dashi. Cook until the sauce is reduced to a few tablespoonfuls.
Garnish with sliced scallions and toasted sesame seeds (or add the garnishes after assembly).
To assemble the gyudon, ladle cooked rice in bowls. Top with the beef and egg. Spoon some sauce over the beef. Serve at once.