Heat 1 tablespoon of butter in a skillet over medium heat. Sauté the chopped onions until translucent and golden. Remove from heat and let them cool completely.
In a small bowl, mix the panko breadcrumbs and milk. Let sit for 5 minutes.
In a large bowl, combine the ground beef, cooled onions, panko-milk mixture, beaten egg, salt, pepper, and nutmeg. Knead the mixture with your hand until it becomes pale and sticky.
Divide the mixture into 4 oval patties. Toss each patty back and forth between your hands about 10 times to release air pockets. Indent the center of each patty with your fingers.
Boil the potatoes in salted water until tender, then mash with butter and milk. Set aside.
Heat a large skillet over medium-high heat with a little oil. Sear the patties for 3 minutes per side until browned.
Add the red wine to the pan, cover with a lid, and steam-cook on low heat for 5-7 minutes until the meat is cooked through.
Remove patties from the pan. In the same pan with the juices, add the remaining butter and mushrooms. Sauté until soft.
Stir in the ketchup and Worcestershire sauce. Simmer until the gravy thickens slightly.
In a separate pan, quickly sauté the bell peppers until blistered but crisp.
Serve the Hambagu steaks topped with the mushroom gravy, alongside the mashed potatoes and sautéed peppers.