How to Make It
In a large, deep skillet, heat 2 tablespoons of the olive oil. Season the quail with salt and pepper, add them to the skillet and cook over moderately high heat, turning once, until browned, about 10 minutes. Transfer the quail to a platter. Add the onions, carrots, garlic, bay leaf, rosemary, thyme and the remaining 1 cup of olive oil, and cook over moderately high heat, stirring occasionally, until softened, about 10 minutes. Add the paprika and cook for 1 minute, stirring. Add the wine and vinegar, season with salt and pepper and bring to a boil.
Return the quail to the skillet, breast side down. Cover and cook over moderately low heat until the meat is very tender and slightly pulling away from the breast bone, about 1 hour. Turn the quail and cook for 10 minutes, uncovered. Let cool for 30 minutes.
Pull the legs from the quail, then pull the meat from the breast bone and discard the backbone and breastbone. Transfer the meat to shallow bowls. Season the sauce with salt and pepper and discard the bay leaf, rosemary and thyme sprigs. Pour the sauce over the quail and serve warm or at room temperature.