Step 1 Heat the oven to 350 degrees. Combine the mangoes and rum in a bowl and mix well. Let stand while you make the rest of the cake.
Step 2 Melt 4 tablespoons butter in a heavy, 10-inch oven-proof skillet, set over low heat. Whisk in the brown sugar until smooth. Remove from the heat.
Step 3Arrange the mango slices evenly in the pan (as you would apples for a tarte Tatin). Pour the remaining juices evenly over.
Step 4Combine the flour, baking powder, salt and lime zest in a small bowl and whisk to blend.
Step 5Beat the one-half cup softened butter with the sugar until smooth. Beat in the egg yolks one at a time. With the mixer on low speed, beat in half the flour mixture, then half the half-and-half, then the remaining flour mixture and the remaining half-and-half. Beat until smooth. Spoon over the mangoes in the skillet, spreading the batter evenly to the edges.
Step 6Bake in the lower third of the oven for 35 to 40 minutes, until the cake tests done. Immediately invert the skillet over a serving plate. Serve the cake warm, with whipped cream, if desired.