1. Preheat the oven to 325°. Sprinkle 2 teaspoons each of the hot and sweet paprika on the short ribs and rub it in well. Season the meat with salt and pepper.
2. In a large enameled cast-iron casserole, heat 1 1/2 tablespoons of the vegetable oil until shimmering. Add half of the short ribs and cook slowly over moderately low heat, turning, until browned, about 10 minutes. Transfer the short ribs to a platter and repeat with the remaining oil and ribs.
3. Add the onions to the casserole and cook over low heat, stirring occasionally, until softened, about 8 minutes. Add the garlic, 1/4 teaspoon of the caraway seeds and the remaining 1 teaspoon each of hot and sweet paprika. Cook over low heat until fragrant, about 4 minutes. Add the Cognac and simmer for 2 minutes. Whisk in the flour and cook for 3 minutes. Increase the heat to moderately high and slowly whisk in the beef stock until smooth.
4. Return the short ribs to the casserole with any accumulated juices and bring to a simmer. Cover the casserole with wax paper and then the lid and bake the stroganoff in the oven for about 2 hours, or until the meat is very tender.
5. Meanwhile, in a small skillet, toast the remaining 1/2 teaspoon of caraway seeds over moderate heat until fragrant, about 30 seconds. Transfer to a plate to cool, then finely grind in a spice grinder or mortar.
6. Return the casserole to the stove over low heat. Stir in the cornichons and toasted caraway. In a bowl, whisk 1/2 cup of the hot cooking liquid into the sour cream. Stir the sauce back into the stroganoff and turn off the heat. Continue stirring until well blended. Serve hot, with the Hand-Cut Spaetzle and Caramelized Carrot Mash.
Make Ahead: The stroganoff can be refrigerated overnight. Rewarm and add the sour cream just before serving.
Wine Recommendation: A spicy, fruity full-bodied Rhône blend would underscore the richness of the meat but also stand up to the other sweet and sour flavors. Look for the 1997 Jean-Luc Colombo Côtes-du-Rhône Les Abeilles or the 1996 Domaine de la RoquetteChâteauneuf-du-Pape.