Wash and dry mangoes. Remove 2 flat sides of each mango with a sharp knife, cutting lengthwise alongside pit and as close to pit as possible, so that mango flesh is in 2 large pieces (reserve remaining fruit for another use).
Make crosshatch pattern with a small, sharp knife, cutting across fruit down to skin at 1cm 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 aluminium foil and sprinkle evenly with 4 tablespoons sugar mixed with cinnamon(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 2 tablespoons 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.