Cut mangoes into 3- to 4-inch-long, 1-inch-wide strips.
Melt 1/2 cup butter in a medium saucepan over medium heat. Add mangoes and brown sugar. Cook, stirring occasionally, 5 minutes or just until mangoes are tender and sugar is melted.
Remove mangoes from saucepan using a slotted spoon. Cook sugar mixture, stirring constantly, 3 more minutes or until reduced to a thick syrup. Remove saucepan from heat, and stir in 2 tablespoons rum.
Arrange mango slices in a spoke design in a single layer in a lightly greased 9-inch round cake pan. Add 1/2 cup sugar syrup to cover mango slices, reserving remaining syrup.
Beat 1/3 cup softened butter at medium speed with an electric mixer until creamy. Gradually add granulated sugar, beating until fluffy. Add eggs, vanilla, and 1 teaspoon rum, beating until blended. Combine flour, baking powder, and salt; add to butter mixture alternately with milk, beginning and ending with flour mixture, beating on low speed until smooth. Carefully spread batter over fruit. (Batter will be very thick.)
Bake at 350° for 30 to 35 minutes or until golden and a long wooden pick inserted into center comes out clean. Cool in pan on a wire rack 30 minutes. Invert onto a cake plate; remove pan. Pour reserved sugar syrup evenly over cake.