Beat the eggs with the sugar and vanilla until very, very frothy and pale. Add the warm milk, melted butter, flour and baking powder, and mix well. The dough should be soft and airy.
Leave to rest for 1 hour at room temperature.
Preheat oven to 180°C.
Butter small individual moulds. In my case, I used small muffin tins. Stir the dough a little more and pour into the molds. Leave 1 cm between the dough and the edges of the moulds.
Bake for 20 minutes.
While the babas are baking, prepare the syrup. Pour the sugar and water into a saucepan and bring to the boil, then lower the heat and simmer for 15 minutes. Remove from the heat and add the rum. Set aside.
When the babas have finished cooking, remove from the moulds and place on a wire rack for 5 minutes.
Place in small containers and drizzle with syrup. Chill before serving with whipped cream and fruit.
