In a large Dutch oven, heat the oil over medium-high heat. Working in 2 batches, sear the short ribs on all sides, 8 to 10 minutes per batch. Transfer to a plate.
To the Dutch oven, add the garlic, scallion whites, kaffir lime leaves, dried chilies and onion. Cook until lightly golden and aromatic, 5 to 6 minutes. Add the short ribs back to the pot with the coconut milk, beef stock, soy sauce and vinegar, and bring to a simmer. Cook, covered, until the short ribs are extremely tender, 3 hours.
Carefully transfer the short ribs to a plate and skim the sauce to remove excess fat. Bring to a boil and cook until lightly reduced, 15 minutes. Add the mushrooms and the short ribs and cook for 1 minute more, then remove from the heat.
Serve over brown rice, garnishing with the sliced scallion greens.