Beat all of the eggs with 6 tablespoons. sugar and milk, adding lemon rind, rum and vanilla.
In a separate pot, add the sugar cubes over medium heat cooking until the sugar takes on a honey color and texture. This takes about 5 minutes and while you should swirl the pan, do not stir as it will possibly contaminate the syrup with sugar crystals. Pour the caramelized sugar cubes into an oven safe pudding form before adding the pudding mixture. A glass pie plate also works.
In a second pot bring enough water to reach up to over the halfway point of the form to a boil and put the filled pudding form into a water bath.
Preheat oven to 350-375 degrees. If you are using a shallow dish, use the lower temperature. Cook for 45 minute in the water bath, and before the Rožata is finished cooking and remove it from the water bath. Cook an additional 15 minutes.
After cooling off, carefully turn the Rožata over onto a plate and serve. In summer, one can refrigerate before serving.