Start making the dough by placing the flour and sugar in a large bowl. Place the yeast into a small jug with the warm milk and mix until the yeast has dissolved and it begins to bubble. Beat the eggs and add everything into the flour mix and combined together. Knead the dough on a worktop until it's smooth and glossy, this will take about 10 minutes. At this point add in the softened butter and work it through the dough thoroughly until it's silky and stretchy.
Place the dough back into a bowl and cover with cling film to prove for at least an hour, until doubled in size.
Turn the dough out of the bowl, and knock it back by kneading it a few times. Grease the rum baba tins so they are ready to go then place the dough into a piping bag and evenly pipe circles into the greased tin. When all of the holes are filled leave in a warm place to proe again, this time for about 20 minutes and preheat the oven to 180°C. Bake in the preheated oven for about 20 minutes, until starting to go golden and cooked through.
Meanwhile make the syrup by putting the sugar and rum in a small saucepan with the water and bring to a boil. Add in the remaining ingredients and let the syrup cook down and infuse. 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, placing them on a suitable surface, and pouring over half the syrup. Allow them to soak up all of the liquid then add the rest. Whilst you are waiting whisk the double cream and vanilla into peaks. Plate up the babas and dollop the cream into the middle.