Remove any silver skin from the surface of the short ribs, and then make 2 cuts across the grain, through the meat down to the bone. (See how-to photo.) Put the ribs in the slow cooker.
Whisk together the broth, soy, sugar, ginger, chili paste, sesame oil, garlic, and scallion whites. Pour over the ribs. Cover and cook on HIGH for 6 hours.
About 30 minutes before serving, put the rice noodles in a bowl with hot water to cover. Soak until tender and pliable, about 30 minutes.
Skim any fat that may collect on top of the beef broth and discard.
Divide the noodles among 6 large shallow bowls and ladle the broth over top. (Cut the meat from the ribs if desired.) Divide the meat evenly among the bowls. Garnish as desired with the carrots, cucumber, cilantro, lime and the remaining green parts of the scallion and more chili paste, if desired.