How to Make It
In a medium skillet, heat the olive oil. Add the onion and rosemary and cook over moderate heat, stirring occasionally, until the onion begins to brown, about 6 minutes. Add the garlic and cook until softened and fragrant, about 5 minutes. Stir in all of the remaining barbecue sauce ingredients and season with salt and pepper. In a food processor, puree the barbecue sauce until smooth. Transfer the sauce to a bowl.
Preheat the oven to 400°. In a medium bowl, toss the shrimp with the olive oil and season with salt and pepper. Wrap each shrimp in a half slice of prosciutto and transfer to a rimmed baking sheet. Brush the shrimp with some of the barbecue sauce and arrange them 1 inch apart. Roast the shrimp for 5 minutes. Turn the shrimp over, brush with more barbecue sauce and roast for about 5 minutes longer, until the shrimp are firm and white throughout and the barbecue sauce is shiny. Transfer the shrimp to a platter and sprinkle with the chopped parsley. Serve with lime wedges and any remaining barbecue sauce for dipping.