Cut the pineapple in half lengthwise from the very top to the bottom (keeping the leaves on). Then cut each half lengthwise in half again, so you have 4 long pieces with leaves at one end.
Add the jelly, brown sugar, butter, cinnamon and salt to a medium saucepan over medium-low heat. Cook, stirring, until the mixture is fully combined and liquid.
Brush the pineapple with some of the mixture (reserve the remaining mixture), transfer to the grill and grill 3 to 4 minutes per side. Remove to a platter and, using a sharp knife, cut the bottom ends off each piece of pineapple. Then, carefully cut lengthwise from the bottom to the top of each piece along the skin (separating the fruit from the skin), and then crosswise into 1-inch pieces.
Drizzle the remaining apricot mixture over the pineapple. Serve with vanilla ice cream.