Prepare the vegetablesTrim the ends of the green beans and cut them into 2-inch pieces. Slice the onion into thin strips (about ¼ inch thick). Then peel the carrot and cut it into bite-sized chunks (½ to 1 inch). Peel the potatoes and cut them into 1-inch wedges.
Combine the seasoningsIn a large pot or Dutch oven, combine sake, mirin, sugar, soy sauce, and water. Stir well to dissolve the sugar, but don’t turn on the heat yet.
Add the beef and onionAdd the shaved beef to the pot, gently loosening the slices as you place them in so they don’t clump together when cooked. Then, add the sliced onions on top.
Start cookingTurn the heat to medium and cook for 3–5 minutes, or until the beef is mostly cooked. Occasionally skim off any foam or impurities that rise to the surface — this helps create a clean, smooth flavor.
Add the potatoes and carrots to the pot. Stir gently to coat everything in the sauce, making sure the vegetables are submerged. Cover with a lid and cook over medium heat for about 10 minutes.
Check doneness and simmerOpen the lid and check if the potatoes are tender by piercing them with a fork. Once they’re soft, lower the heat to medium-low and continue cooking for another 5 minutes with a drop lid (otoshibuta).
Add green beans and finish cookingAdd the green beans and cook over medium heat for 2–3 minutes, or until they turn bright green and tender.
Serve and enjoyServe warm, ideally with a bowl of steamed rice.