Season beef with cayenne pepper, kosher salt, and pepper.
Heat peanut oil in a pan over high heat. Add beef and brown for 3 minutes on each side. Turn off the heat and transfer beef to a plate.
Let the pan cool slightly, 1 to 2 minutes. Add butter and melt butter in the residual heat. Turn the heat to medium and add mushrooms, cut-sides down. Cook until mushrooms are light brown, 3 to 4 minutes. Flip and continue to cook until golden, 2 to 3 minutes more. Add onion and season with salt; cook onion until slightly golden, 3 to 4 minutes.
Pour sake into the pan and bring to a boil while scraping the browned bits of food off the bottom of the pan with a wooden spoon. Stir in chicken broth, miso, maple syrup, and ketchup, and bring to a simmer. Add browned beef, mushrooms, and any accumulated mushroom juices.
Reduce the heat to low or medium-low, cover, and simmer until meat is fork-tender, 2 to 3 hours, turning beef halfway through cooking time.
Stir in green onions, chile pepper, and vinegar. Cover and cook for 5 minutes more. Taste and adjust seasonings.
