Pour condensed milk into a 9-inch deep-dish pie plate and cover with foil. Put pie plate in a roasting pan and add enough hot water to pan to reach halfway up side of pie plate. Bake milk in middle of oven until thick and golden, about 1 1/2 hours. Carefully remove pie plate from pan and cool completely.
Add rum and vanilla to milk, whisk until smooth, and season with salt. (The dulce de leche may be made 2 days ahead and chilled, covered.)
Prepare a grill or grill pan. Cut the pineapple lengthwise through leaves, keeping leaves attached, into 4 wedges. Grill the pineapple wedges, cut-side down, until just charred, about 2 minutes on each cut side.
Divide the wedges among plates, put a scoop of ice cream next to each, and drizzle with the dulce de leche.