1.Warm milk in a small saucepan. Whisk yeast, warmed milk and sugar in a large bowl until combined. Cover; stand in a warm place about 10 minutes or until frothy.
2.Add sifted flour and eggs to yeast mixture; mix to a soft dough. Knead dough on a floured surface about 10 minutes or until smooth and elastic. Place in a lightly oiled large bowl, cover; stand in a warm place about 1 hour or until doubled in size.
3.Grease four 12-hole (1-tablespoon/20ml) mini muffin pans.
4.Mix butter and sultanas into dough; knead on a floured surface until combined. Roll 1 rounded teaspoon of dough into a ball and place each in a pan hole. Cover pans; stand in a warm place for about 1 hour or until doubled in size.
5.Preheat oven to 160°C/320°F.
6.Bake babas about 20 minutes or until browned lightly; cool in pans.
7.Meanwhile, to make rum syrup, stir the water, sugar, vanilla bean, cinnamon, lime and orange in a large saucepan over medium heat, without boiling, until sugar dissolves. Bring to the boil. Reduce heat; simmer, uncovered, without stirring, 8 minutes. Remove from heat; stir in rum.
8.Place babas in a large heatproof bowl; pour over hot syrup. Stand at room temperature 1 hour for syrup to soak in; refrigerate 1 hour before serving.