Summarize this article:

Use a sharp knife to shave the beef into very thin slices. Cut the onion and shiitake mushrooms into moderately thin slices, as well.  To save yourself time and effort, ask the butcher to slice the beef very thinly before packaging it. If the butcher does not slice the beef, freeze it for 1 hour before slicing it yourself. Partially frozen meat is easier to cut than fully thawed meat. The slices of beef should be just a little thicker than cold cut lunchmeat is. This thinness is key. Beef that is too thick will not cook quickly enough. The slices of onion and mushroom should both be approximately 1/3 inch (1 cm) thick. Place the butter in a medium saucepan and set it on the stove. Heat over medium for a few minutes, just until the butter is completely melted. Add the sliced onions and sliced shiitake mushrooms to the melted butter. Cook, stirring frequently, for 4 to 5 minutes. The onion should become translucent and the shiitake mushrooms should feel notably tender. Add both alcohols to the saucepan. Cook for another 2 minutes. During this time, most of the actual alcohol should burn away, leaving only the flavor behind. Add the water, Dashi powder, soy sauce, sugar, grated ginger, and minced garlic to the contents of the saucepan. Stir to combine. Allow the contents of the pan to reach a steady simmer before continuing. Place the thin slices of raw beef in the saucepan. Reduce the heat to low and cook gently for 3 to 5 minutes.  Use cooking chopsticks or tongs to carefully separate the beef slices as they cook. Doing so can help prevent the beef from sticking together. This may seem like an abnormally short cooking time, but if the beef has been sliced to an adequately thin size, it should be plenty. Do not overcook the beef since it can easily become dry. Fill two serving bowls with freshly steamed white rice. Divide the gyudon evenly and spread it over the rice in both bowls. For an even more authentic experience, prepare Asian-style sticky rice or sushi rice instead of relying on instant rice. Top each serving of gyudon with an organic raw egg. Crack the egg directly over the beef, allowing the yolk to remain intact in the center of the bowl when you serve the dish.  Please exercise caution when consuming raw eggs. Organic eggs from a clean, reliable source might be safe to eat, but the USDA discourages the consumption of raw eggs altogether due to the risk of salmonella. If you feel squeamish about consuming raw eggs, you may skip this step. If you do opt to include the raw egg, mix it into the beef and rice as you eat the gyudon. Doing so can give the taste of the dish a rich, creamy undertone. Gyudon is frequently topped with a bit of shichimi togarashi and beni shoga. Pair it with sides like miso soup and steamed vegetables. A combination of broccoli, cauliflower, and sliced carrots will pair especially well with this dish, but you can serve it with any vegetables you prefer. Consider steaming the vegetables for a few minutes, allowing them to become tender-crisp without turning into mush.

Summary:
Slice the beef and vegetables. Melt the butter. Sauté the onion and shiitake mushrooms. Mix in the sake and mirin. Stir in the water and remaining seasonings. Add the beef and simmer gently. Serve over rice. Top with the egg. Add any other desired accompaniments.