In a medium saucepan , heat the oil. Add the onion and rosemary and cook over moderate heat, stirring occasionally, until the onion begins to brown. Stir in all of the remaining barbecue sauce ingredients and season with salt and pepper. In a food processor, purée the barbecue sauce until smooth. Transfer the sauce to a bowl.

Preheat the oven to 400°. Coat shrimp with barbecue sauce. Wrap each shrimp in a half a 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 to 6 minutes. Turn the shrimp over, brush with more barbecue sauce and roast for an additional five minutes longer. Broil for 1 to 1 1/2 minutes to get a nice golden crispness. Transfer the shrimp to a platter and serve. Garnish with lime wedges and any remaining barbecue sauce for dipping.