Season the short ribs with salt and pepper on all sides. Heat a skillet over medium-high heat and brown the ribs in batches until a crust forms, then transfer them to the slow cooker.
Add the sliced onion, carrot chunks, half the minced garlic, thyme sprigs, rosemary sprig, red wine, and beef broth to the slow cooker. Stir gently to combine the liquids and vegetables around the ribs.
Cover and cook on low for 8 hours until the meat is very tender and pulls away from the bone easily.
About 30 minutes before the ribs finish, place the quartered potatoes in a large pot and cover with cold water. Bring to a boil and cook until the potatoes are soft when pierced with a fork.
Drain the potatoes and return them to the pot. Add the remaining minced garlic, butter, milk, and a pinch of salt. Mash until smooth and creamy.
Remove the herb sprigs from the slow cooker. Serve the short ribs and some of the cooking liquid over the garlic mashed potatoes.