Special equipment: a 9 1/2-by-2-inch round baking pan
For the dough: In a large mixing bowl, add the butter, sugar and salt. Using your hands, crush and combine all the ingredients well. Add the 3 whole eggs and 1 yolk, reserving the remaining egg white. Combine with your hands once again.
Add the flour and yeast and continue to mix with your hands until all the ingredients are combined, about 2 minutes. Scrape the dough onto a work surface and knead until a smooth dough is formed. Wrap the dough in plastic wrap, and allow to rest at least 6 hours or up to overnight.
For the custard: In a medium saucepan, heat the milk until it begins to bubble at the edge, then remove from the heat.
Meanwhile, in a mixing bowl, combine the 3 eggs with the sugar and whisk until the sugar has dissolved. Add the flour and whisk for a few more minutes until smooth and thickened.
Slowly pour 2/3 of the hot milk into the egg mixture and whisk, then pour the mixture back into the saucepan and return to the heat, whisking, until thickened and a custard is formed. Transfer the custard to a mixing bowl, then fold in the rum. Allow to cool.
Preheat the oven to 350 degrees F. Take the dough out of the fridge, cut it in half, then place it onto a floured work surface. Knead the dough for a minute to soften, then using a rolling pin, flatten the dough to a 1/5-inch-thick round base. Now using a 9 1/2 inch round ring, cut the dough with a small knife.
Oil a 9 1/2-by-2-inch round baking pan, then place the cut dough inside. Pour the custard on top of the dough, leaving 1/3-inch from the edge empty.
Place the remaining dough onto the floured work surface and roll into the same 1/5-inch-thick round base. Gently place the dough over the custard and press lightly to seal the edges.
Brush the top with the reserved egg white, then using a fork, lightly make a criss-cross pattern over the surface of the cake.
Bake the cake until golden brown, about 40 minutes.
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