Preheat the oven to 350 degrees F. Grease an 8-inch square baking pan with vegetable oil and line the bottom with parchment paper.
Make the cake: Beat the eggs and sugar in a large bowl with a mixer until pale and slightly thick, about 6 minutes. Slowly beat in the melted butter, then beat in the vanilla and 1/4 cup whole milk.
Whisk the flour, baking powder and salt in a bowl. Sift the flour mixture over the egg mixture, then gently fold together with a rubber spatula to make a thick batter. Transfer the batter to the prepared pan and bake until a toothpick inserted into the center of the cake comes out clean, about 30 minutes. Run a knife along the edges to release the cake from the pan.
Meanwhile, whisk the remaining 2 cups whole milk, the evaporated milk, condensed milk and rum in a bowl. Pierce the warm cake all over with a fork, then pour the milk mixture gradually and evenly on top, letting the cake absorb the liquid until all is used. Cover and refrigerate at least 1 hour or overnight.
Before serving, make the topping: Peel, pit and dice the mangoes. Puree half the mangoes in a blender with 2 tablespoons sugar. Transfer to a bowl and fold in the remaining diced mango.In a separate bowl, beat the heavy cream and the remaining 1 tablespoon sugar with a mixer until soft peaks form. Slice the cake; top each piece with the whipped cream and sprinkle with cinnamon. Serve the mango topping on the side.