Preheat the oven to 325°F (165°C).
Pat the beef short ribs dry and season generously with kosher salt and black pepper. Sprinkle with flour and turn the ribs until lightly coated on all sides.
Heat a large oven-safe Dutch oven over high heat. Working in batches if necessary, sear the short ribs until deeply browned on both sides. Transfer the browned ribs to a plate and carefully discard excess grease from the pot.
Reduce the heat to medium. Add the butter and sliced onions. Cook for about 5 minutes, stirring occasionally, until the onions begin to soften.
Pour 1/2 cup of the apple cider over the onions. Season lightly with salt and pepper and continue cooking for 5–8 minutes, stirring occasionally, until most of the cider has reduced and the onions become soft and lightly caramelized.
Stir in the fresh thyme. Return the short ribs to the Dutch oven.
Pour in the remaining 1 1/2 cups apple cider and the broth. Add the carrots, smashed garlic, and bay leaves. If using the apple, arrange the quarters around the beef.
Cover the Dutch oven and transfer it to the oven. Braise the Cider Braised Short Ribs for 2 1/2–3 hours, or until the beef is very tender and easily pulls away from the bone.
Remove and discard the bay leaves. Taste the pan sauce and adjust the salt and black pepper as needed.
Serve the short ribs over mashed potatoes with the caramelized onions, carrots, and generous spoonfuls of the cider-infused pan sauce.