Heat grill to high. Place rum, brown sugar, and salt in a small saucepan and cook until the sugar has melted and the mixture reduces and thickens slightly. Stir in butter until melted. Remove from the heat.
Brush plantains with oil on both sides and grill until golden brown and caramelized, about 3 to 4 minutes per side. Brush with the glaze during the last few minutes of grilling. Remove from the grill and brush with more of the glaze before serving.