If you like more of a kick, add extra minced chile to the meat and sauce in step 7 and simmer briefly. You'll need a spice grinder or clean coffee grinder for grinding the cardamom pods and fennel seeds.
Prepare ribs: Heat a 12-in. frying pan (not nonstick) over medium-high heat. Meanwhile, season short ribs all over with 1 1/4 tsp. salt. Add oil to pan and heat until hot. Brown half of ribs at a time, turning as needed to brown on all sides, 7 to 10 minutes per batch. Transfer as done to a rimmed baking sheet or platter. Meanwhile, preheat oven to 325°.
Discard all but 2 tbsp. fat from pan and reduce heat to medium. Add onion and 1/2 tsp. salt. Cook, stirring often and scraping up caramelized bits from bottom of pan, until onion is soft, 6 to 8 minutes. Add ginger and chiles and cook, stirring, until softened, 3 to 5 minutes.
Add spices and continue to cook and stir until they start to stick to bottom of pan, 30 to 60 seconds. Stir in tomatoes, breaking up with side of spoon, and bring to a simmer. Stir in remaining 1/2 tsp. salt.
Put ribs in a single layer in a wide 6- to 8-qt. dutch oven or other ovenproof pot and scrape sauce out of pan and over ribs. Cover pot and bake ribs for 45 minutes. If needed, add 2 cups hot water so there's about 1/2 in. liquid in pan. Continue to cook, covered, turning once and adding more water if it gets below 1 in., until meat comes away easily from bone, about 3 1/4 more hours.
Meanwhile, prepare raita: Toast cumin seeds in a small frying pan over medium heat, stirring, until fragrant and a shade darker, about 2 minutes. Grind. Combine cumin with remaining raita ingredients, including salt. Chill.
Transfer ribs from sauce to a rimmed baking sheet and let cool.
Discard bones, fat, and gristle from ribs and tear meat into large shreds. Let sauce cool, then skim off fat. Measure sauce and add water or boil down, stirring often, to make 3 cups. Stir meat into sauce and season with additional salt to taste. Keep warm over low heat.
Preheat broiler with rack 3 to 4 in. from heat. Set rolls cut side up on rimmed baking sheets. Brush with ghee. Broil until golden, 1 to 2 minutes.
Spoon about 1/4 cup short rib mixture over bottom of each bun. Top each with about 1 tbsp. raita and the bun top.
*Find garam masala (a blend of warm Indian spices), black cardamom pods, and ghee (clarified butter) at well-stocked markets, Indian markets, and on amazon.com; the spices are also sold at worldspice.com. Crack cardamom with a mallet or hammer. Grind 2 to 3 minutes in a spice grinder, then discard any fibrous bits that haven't broken down.
Make ahead: Through step 6 up to 1 day, chilled.
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.