Preheat oven to 350°F (175°C). Grease a 9x13 inch baking dish with butter and dust lightly with flour.

In a large mixing bowl, beat 6 eggs with sugar using a hand mixer on high speed for 8-10 minutes until thick, pale, and tripled in volume.

Sift flour, baking powder, and salt together. Gently fold the dry ingredients into the egg mixture in three additions, being careful not to deflate the batter.

Fold in vanilla extract and melted butter until just combined.

Pour batter into prepared pan and bake for 25-30 minutes until golden and a toothpick inserted in center comes out clean.

While cake bakes, prepare the rum syrup: combine water, sugar, and rum in a saucepan and simmer for 5 minutes. Set aside to cool.

For the custard, heat milk in a saucepan until just simmering. In a bowl, whisk egg yolks with sugar until pale.

Gradually whisk hot milk into egg mixture, then return to saucepan. Cook over medium-low heat, stirring constantly, until custard thickens enough to coat the back of a spoon (about 8-10 minutes). Remove from heat and stir in vanilla.

When cake is done, poke holes all over with a fork and slowly pour rum syrup over the warm cake, allowing it to absorb completely.

Pour custard evenly over the soaked cake and spread gently with a spatula. Refrigerate for at least 2 hours.

For caramel topping, melt sugar in a heavy saucepan over medium heat without stirring. Once melted and amber colored, carefully add butter and cream, stirring until smooth.

Let caramel cool for 5 minutes, then pour over chilled custard layer. Refrigerate until caramel sets, about 30 minutes before serving.