In a 10- to 12-inch frying pan, crush cardamom seed and lemon peel with back of a spoon. Add sugar and 1 cup water. Stir over high heat until sugar dissolves. Boil gently, uncovered, until syrup is reduced to 1/2 cup, 8 to 10 minutes.
Meanwhile, cut peel off mangoes, discard pits, and cut fruit into 1/2-inch-wide slices, about 3 inches long. If making ahead, cover fruit and syrup and set aside up to 4 hours.
Add mangoes to hot syrup and cook over high heat, turning once, until fruit is hot, 2 to 3 minutes. In a small pan (about 1-qt. size), warm rum until hot. Ignite rum and add to mangoes (not beneath a vent, fan, or flammables). Spoon mangoes and flaming syrup over frozen yogurt in bowls.