Trim the bottom 1 to 2 inches off the enoki mushrooms and rinse them under cold running water. Separate the mushrooms into small bundles, about 1/4 inch thick.
Lay one slice of beef flat on a cutting board. Place a mushroom bundle at one end and roll the beef tightly around it. Repeat this process until all the beef and mushrooms are used.
In a small bowl, whisk together the water, soy sauce, sake, mirin, and minced garlic to make the sauce.
Heat the avocado oil in a large skillet over medium-high heat. Add the sliced onions and cook for about 3 minutes, stirring often, until they begin to soften.
Arrange the beef and mushroom rolls in a single layer on top of the onions. Pour the sauce mixture evenly over the rolls and place a lid on the skillet.
Bring the liquid to a boil, then lower the heat to a simmer. Cook for 3 to 4 minutes until the beef reaches a safe internal temperature of 145°F (63°C).
Remove the lid and pour the whisked eggs over the beef rolls. Cover the skillet again and cook for 1 more minute until the eggs are set and reach 160°F (71°C).
Remove the skillet from the heat. Let the beef rest for 3 minutes for food safety. Sprinkle with green onions and sesame seeds before serving.