Place the flour in a large bowl. Place the yeast on one side of the bowl and the salt on the other side. (Make sure the salt is not placed on top of the yeast, as it can kill it, making it inactive). Add the sugar and stir everything together with a spoon until evenly mixed.
Mix together the milk and eggs until well combined.
Add three-quarters of the combined eggs and milk to the flour and stir to combine.
Mix in the rest of the liquid and knead the dough on a worktop until it’s smooth and glossy, this will take approximately 10 minutes.
Add in the softened butter and work it through the dough thoroughly until it’s silky and stretchy. This should take approximately six minutes.
Place the dough back into a bowl and cover with cling film. Set the dough aside to rise for at least an hour, until doubled in size.
Grease and sugar the four 11cm/4½in fluted rum baba tins (savarin moulds). (Adding the sugar will help the fragile sponges come out of the moulds).
Turn the dough out of the bowl, and knock it back by kneading it a few times.
Place the dough into a piping bag with a large plain nozzle. Pipe the dough into the four moulds. Try and get them all as equal as possible.
Preheat the oven to 180C/350F/Gas 4.
Allow to prove for a second time until the dough has expanded almost to the top. Be careful not to over-prove at this stage, or you will get a muffin top around the edges.
Bake in the preheated oven for about 20-25 minutes.
Meanwhile for the syrup, put the sugar and rum in a small saucepan with 200ml/7fl oz water and bring to a rolling boil.
When the babas are cooked, take them out of the oven and allow to cool a little before carefully removing the cakes from their tins. They will be very fragile.
Place the babas onto a dish and pour over half the syrup. Allow them to soak up all of the liquid; then turn them over and repeat with the rest of the liquid. Transfer to the fridge to chill.
Meanwhile for the Chantilly cream, whip the cream with the icing sugar and vanilla seeds. The cream must be firm enough to pipe and hold its shape on top of the babas. Transfer the cream to a piping bag and keep in the fridge until needed. Prepare the fruit as necessary.
To serve, pipe the Chantilly cream, using a star nozzle, into the middle of the babas. Garnish with the mixed fruit.