Heat the oil in a small saucepan over medium heat. Add the onion and cook, stirring occasionally, until softened, about 8 minutes. Add the paprika and cook, stirring, 30 seconds. Add the cranberry sauce, ketchup, sugar, vinegar, Worcestershire and 1/4 cup water. Bring to boil over medium-high heat, then reduce the heat slightly and simmer until slightly thickened and the flavors have blended, about 10 minutes. Let cool slightly, then transfer to a blender or food processor (be careful when blending hot liquids) and blend until smooth. Transfer to a bowl and set aside.
Put the dressing in a large bowl with the parsley and egg and mix well with your hands. Divide the mixture into 6 portions and form each into a flattened 3-inch cake. Put the cakes on a plate and refrigerate to set, about 10 minutes.
Melt the butter over medium heat in a skillet. Add the dressing cakes and cook, turning once, until hot and golden brown on both sides, 3 to 4 minutes per side. Transfer the cakes to individual serving plates. Add the broth to the skillet, then the sliced turkey and cook gently until just warmed. Serve the cakes topped with the turkey and cranberry barbecue sauce.
Use 1 to 2 eggs, depending on how dry your stuffing is.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.