Step 1: Prepare the shrimp and marinade: Rinse the shrimp under cold running water, then drain them and blot dry with paper towels. Using a the tip of a paring knife, make two “starter” holes — one at the top and one at the bottom — in each shrimp.
Step 2: Prepare the sugarcane skewers: Using a long sharp knife, cut the sugarcane swizzle sticks into 3-inch lengths, slicing them sharply on the diagonal to make a sharp point. You will need one skewer for each shrimp. Thread the shrimp onto the sugarcane skewers, one to a skewer. Arrange in a foil pan and season with salt, pepper, olive oil, and lime juice, turning the shrimp to coat both sides.
Step 3: Make the glaze: Combine the jerk seasoning, butter, rum, brown sugar, and lime juice in a heavy saucepan. Boil over medium-high heat until thick and syrupy, whisking often.
Step 4: Set up the grill for direct grilling and preheat to high. When ready to cook, brush and oil the grill grate. Arrange the shrimp on the hot grate and grill, turning with tongs, until nicely browned on the outside and firm and pink inside, about 2 minutes per side. Brush the shrimp with the glaze as they cook.
Step 5: Transfer the shrimp to serving plates or a platter. Spoon any remaining rum glaze over the shrimp and serve at once.
Sugarcane swizzle sticks can be purchased at specialty markets or through www.melissas.com.