Grind sesame seeds to a coarse powder in grinder. Reserve 1/4 cup scallion greens, then whisk together remaining scallions, garlic, soy sauce, brown sugar, hot-pepper paste, sesame oil, and 2 tablespoons sesame-seed powder in a large bowl. Reserve remaining sesame-seed powder for serving. Add short ribs to soy sauce mixture, rubbing mixture into them. Transfer ribs to a large sealable plastic bag and seal bag, pressing out excess air. Marinate, chilled, at least 8 hours.
Transfer ribs to a wide 6- to 8-quart heavy pot and add water and ginger. Simmer, tightly covered, until ribs are very tender, about 3 hours.
Transfer ribs to a platter using tongs and keep warm, covered with foil. Skim fat from sauce and pour sauce through a sieve lined with a dampened paper towel into a bowl, then discard solids. Serve ribs with sauce in shallow bowls and sprinkle with reserved scallion greens and remaining sesame-seed powder.
• Ribs can be marinated up to 1 day.
• Ribs can be braised 2 days ahead and cooled in sauce, uncovered, then chilled, covered. Remove fat from sauce, then reheat, covered, over moderate heat and proceed with recipe.