Wash and dry mangoes. Remove 2 flat sides of each mango with a sharp knife, cutting lengthwise alongside pit and cutting as close to pit as possible so that mango flesh is in 2 large pieces (reserve remaining fruit for another use). Make a crosshatch pattern with a small sharp knife, cutting across fruit down to skin at 1/2-inch intervals and being careful not to pierce through. Grasp fruit at both ends and turn inside out to make flesh side convex.
Arrange fruit, skin side down, in a large shallow baking pan lined with foil and sprinkle evenly with 4 tablespoons turbinado sugar (total). Broil 5 inches from heat until fruit is golden brown (it will not brown evenly), about 5 minutes. Arrange fruit on a large platter.
Cook rum with remaining sugar in a small saucepan over moderately low heat, stirring, until sugar is dissolved. Remove from heat, then carefully ignite rum with a kitchen match and pour, still flaming, over warm mangoes. Serve immediately.