Place the oven rack in the middle of the oven. Preheat the oven to 350 degrees F. Spread the butter into the bottom of a 9", spring-form cake pan.
Combine the brown sugar and spices in a small bowl and then sprinkle into the cake pan. Fan the slices of mango in a circular pattern on top of the brown sugar.
Place flour, baking powder, salt and lime zest in a medium bowl and whisk to combine. Place 1/2 cup butter, granulated sugar and vanilla in a large bowl and beat with an electric mixer until well combined and lightened. Beat in the eggs one at a time. Gradually beat in the flour mixture alternately with milk.
Spoon the batter over the mango. Bake for 45 minutes, or until a cake tester inserted in the centre pulls out clean. Cool the cake for 15 minutes. Run a paring knife around the edge of the cake, unhinge and remove the outer ring of the pan. Carefully invert the cake onto a plate and serve.