Cut the green onions into 3" sections. Separate the whites & the greens.
In a small bowl, crack in 2 eggs and beat until the whites and yolk are somewhat combined.
In another small bowl, mix together the dashi, sake, soy sauce, and sugar.
In a small pan (about the same size as your bowl) on medium heat, pour in half of the sauce and add the half of the white parts of the green onions and bring to a boil.
Once boiling, add in half of the beef and half of the green onion. Cover with a lid and let the beef cook (since the beef is thin it should cook in about a minute).
When the beef is cooked, pour the egg into the pan in a figure 8 shape. Rotate the pan or give the ingredients a quick stir so the egg spreads around more evenly.
Cook until the egg is just slightly runny, about a minute. If you want it to be cooked through, you can cover with a lid and cook longer.
Use a spatula to carefully release the bottom of the beef and egg mixture without breaking it apart (it should release fairly easily). Then slide the whole thing off the pan and into a bowl with rice.
Repeat steps with the leftover ingredients to make a second donburi.