Season beef with kosher salt, black pepper, and cayenne.
Heat peanut oil in a pan over high heat. Brown beef strips on all sides, about 3 minutes per side. Turn off heat and transfer beef to a plate.
Let pan cool slightly, 1 to 2 minutes; melt butter in residual heat. Turn heat to medium and add mushrooms, cut-side down. Cook until mushrooms are lightly brown, 3 to 4 minutes. Flip; continue to cook until golden, 2 to 3 minutes more. Add onion and season with salt; cook until slightly golden, 3 to 4 minutes.
Stir in sake and deglaze the pan, scraping up browned bits using a wooden spoon. Stir in miso, maple syrup, ketchup, and chicken broth; bring to a simmer, about 5 minutes. Add browned beef, mushrooms, and their juices.
Cover and let simmer over low to medium-low heat until meat is fork-tender, 2 to 3 hours, turning beef halfway through cooking time.
Stir in vinegar, chiles, and green onions. Cover and cook for 5 minutes more. Taste and adjust seasonings.