Trim the base of the enoki mushrooms and separate them into small even bundles, then set aside while you prep the beef.
Lay thin beef slices flat and tightly wrap each piece around a bundle of enoki mushrooms, forming compact rolls.
In a bowl, mix soy sauce, mirin, sake, dashi stock, sugar, garlic, ginger, and water until well combined.
Heat oil in a large skillet over medium heat and sauté sliced onions until slightly softened and fragrant.
Place beef enoki rolls over the onions in a single layer and pour the prepared sauce evenly over them.
Cover and simmer for 3–5 minutes until the beef turns mostly cooked and the mushrooms soften.
Slowly pour whisked eggs over the pan and cook covered for 1–2 minutes until just set and silky.
Remove from heat, garnish with scallions and sesame seeds, and serve hot over steamed rice.