The shrimp are not really barbecued; in New Orleans that means shrimp in butter sauce,  not barbecue sauce!  My brother, Jackie, and his wife, Jane, said this recipe is inspired by Pasqual's Manale restaurant in New Orleans.  They called it "barbecued" because the sauce turns a bit pinkish red  from the spices. It was invented at Pasqual's Manale in 1953, but the restaurant has been around since 1913 and is still open for business.   According to local legend, a customer who had just been to Chicago came into Pasqual's Manale in New Orleans and told the chef about a shrimp recipe he ate in Chicago.  It had shrimp, butter, and lots of pepper.  The Pasqual's Manale chef recreated his own version and it was named "Barbecued Shrimp" because of its smoky flavor from black pepper and Worcestershire sauce, not because it's actually barbecued on a grill! 
The recipe calls for head-on shrimp.  I just compromise and remove the head, leaving the peel and the tail.  They make for great flavor in the sauce and a guaranteed need for lots of napkins to go with the dish.  It's great as an appetizer or main meal, and ALWAYS served with hot French bread for soaking up the delicious sauce.  As a main meal, a salad and roasted potatoes would be great additions.