Pre-heat oven to 350 degrees. Prepare a 9 inch round cake pan by greasing the bottom and sides liberally with the butter or non-dairy substitute. Sprinkle 1 tablespoon of sugar over the bottom of the pan in an even layer. Lay sliced mango on top of the sugar in a nice spiral pattern.
Put rice in a large glass bowl and cover with enough water to come up about 1 inch over the top of the rice. Soak rice for at least 1 hour. Drain off water, rinse with running water and put back into bowl with 1½ cups fresh water. Cover tightly with plastic wrap. Microwave on high for 5 minutes.
Stir rice thoroughly and return to microwave for another 5 minutes. Stir again. Rice should be tender, cooked through and sticky. If not, microwave for another 3 minutes and check again. Let stand covered for 5 minutes when done. Alternately, you can cook the soaked rice in a rice cooker.
While rice is cooking, heat coconut milk with ¼ cup sugar and a small pinch of sea salt until almost boiling over medium heat. Do not cook it over too high of a heat or it will curdle. Cook until all the sugar is dissolved and it is very hot. Pour hot mixture over cooked rice; stir well and let sit for at least 5 minutes.
Melt 4 tablespoons butter or non-dairy substitute over medium heat with ¾ cup of sugar until it forms a thick caramel. Gently pour over the mango slices in the cake pan. Add rice mixture and press down to pack it well in the pan. Smooth out the top with a rubber spatula.
Place cake pan in a sheet pan to catch any caramel that may bubble over. Bake in oven for 20-25 minutes or until the rice is lightly browned and the caramel is hot and bubbling. Let cool for 5 minutes and then run a knife around the pan to loosen. Invert on serving plate and let cool. The more you let it cool, the easier it is to slice.