Season the short ribs all over with salt and pepper. Warm the olive oil in a large skillet or frying pan over medium-high heat. Sear the short rib pieces, turning occasionally, until deep brown all over, about 15 minutes total. Transfer the short ribs to the crock of a slow cooker.
Pour off all but 2 tablespoons of the fat in the skillet. Add the onion, carrots, and garlic to the pan, sprinkle with salt, and cook over medium heat until the vegetables are browned and softened, about 8 minutes. Add the paprika, caraway, and coriander and cook, stirring, until fragrant, about 1 minute.
Transfer the onion-carrot mixture to the slow cooker. Add the broth, thyme, and bay leaf, cover, and bring to a boil on high. Cook for 4 hours, until the meat is very tender. Discard the thyme stems and bay leaf. Spoon off as much of the fat as you can from the surface.
Meanwhile, bring a large pot of heavily salted water to a boil. Cook the egg noodles according to the package directions. Drain, then return them to the cooking pot. Add the butter and toss with tongs until the noodles are coated.
Serve the noodles topped with goulash in warmed deep bowls. Garnish each with 1 tablespoon of the sour cream and serve.