Combine flour, yeast, salt, and sugar in a large bowl, keeping salt and yeast separate.
Whisk the milk and eggs together until combined.
Stir three quarters of the egg mixture into the flour.
Add remaining liquid and knead for 10 minutes until smooth.
Work softened butter into the dough for 6 minutes until silky.
Cover and let rise for one hour until doubled in size.
Grease and sugar four 11cm savarin moulds.
Knock back the dough by kneading it briefly.
Pipe equal amounts of dough into the prepared moulds.
Preheat the oven to 180°C.
Prove a second time until the dough nearly reaches the top of the moulds.
Bake for 20 to 25 minutes.
Boil sugar, rum, and 200ml of water to create a syrup.
Cool slightly and carefully remove babas from the tins.
Soak babas thoroughly in syrup and chill in the fridge.
Whip double cream with icing sugar and vanilla seeds until firm.
Pipe cream into the center of each baba and garnish with fruit.