Preheat oven to 300F. Pat short ribs dry and season all over with salt and pepper. Heat a Dutch oven (or a large oven-safe high-walled skillet) over medium-high heat, then add butter and swirl to evenly coat. Sear short ribs 4-6 minutes on all sides until a deep golden brown. Transfer short ribs to a plate and set aside.
Allow pan to cool off slightly and reduce heat to medium. Add green onions to drippings and cook 1-2 minutes until wilted. Add sliced ginger and garlic and cook an additional 1-2 minutes until fragrant.
Pour soy sauce, mirin, beef broth, and brown sugar into pan and use a wooden spoon to scrape up any browned bits. Bring sauce to a simmer.
Once sauce is simmering, remove pot from heat and nestle short ribs back into pot. Cover pot and place in preheated 300F oven. Roast short ribs 2 to 2-1/2 hours until short ribs are very tender and meat pulls away easily from the bone with a fork.
Remove short ribs from pot and transfer to a plate. Skim any excess fat from surface of liquid and discard. Whisk cornstarch-water mixture into pot liquid until smooth. Place short ribs back in sauce and arrange bok choy in pot.
Cover pot and place back in 300F oven and allow to cook an additional 15-20 minutes until bok choy is tender and sauce is thickened.
Serve short ribs and boy choy warm over steamed rice with extra sauce from pot drizzled over top. Enjoy!