Remove the compact root block from the enoki, rinse the stems gently under cold water, and separate them into 15 evenly sized portions. Keep each portion gathered at one end so the loose strands remain manageable.
Wind one beef slice around each mushroom portion, then put the assembled bundles aside. A snug wrap should gather the stems without squeezing them into a bulky knot.
Stir the dashi, sake, mirin, soy sauce, and sugar together in a bowl, then reserve the mixture. The sugar should dissolve into the liquid rather than collect at the bottom.
Soften the sliced onion in a pan with 2 tsp oil over medium heat for about 4–5 minutes. Look for flexible slices that will form an even bed beneath the rolls.
Arrange the bundles on the onions, add the sauce, and cover the pan for about 5 minutes of simmering. The lid holds in steam, and the mushrooms should soften and draw together slightly as they cook.
Distribute the beaten eggs over the bundles and cover again for an additional 1 minute, or until the topping begins to set. The egg should start turning opaque while retaining a soft appearance.
Allow the rolls to cool briefly, then scatter the chopped scallions over the top. The scallion pieces add a fresh contrast to the warm beef and egg.