First of all, choose a quite big bowl: the dough will rise better. If the bowl of your stand mixer is not big enough, mix the ingredients and them truest the dough into high sided bowl.
Sift in 100g of flour and mix the yeast you have previously dissolved in little lukewarm milk;
Turn on the stand mixer provided with the paddle attachment at 2-3 speed;
Combine for a few minutes;
Cover with a tea-towel and set aside to rest for 30 minutes;
Sift the remaining flour into the stand mixer bowl;
Add the cold (from the fridge) eggs
Add the eggs one by one: don’t be in a hurry, this is an essential passage. Wait until the first egg has been completely incorporated in the dough before adding the following one;
Repeat this five times according to the numbers of the eggs;
Gradually add the sugar with the mix running;
When then sugar has been incorporated, add a pinch of salt (which favors the rising process);
Knead five minutes more;
You should replace the paddle attachment with the hook which will keep the dough elastic and smooth;
Add the softened butter at room temperature: don’t melt it in a pan over the heat, it must keep the so-called creamy texture (you may shortly warm it in the microwave);
Cut the butter into small dices, they will combine with the dough more easily;
Add a few dices of butter a little at a time when the previous ones have been wholly incorporated;
If the dough results too hard and stiff, add some milk until soft and elastic (be careful: do not add too much milk: you can’t make it right again);
Scrap the vanilla pod with a sharp knife and add the vanilla to the dough (you may add the lemon zest here);
Knead for 15 minutes more until a smooth, soft light yellow dough;
Increase the speed of the stand mixer to medium;
Cover the dough with cling film: leave it in the stand mixer bowl or in a bigger one; it will rise quite a lot;
Allow it to rise for 3 hours at 25-26°C (77-79°F). You may also pre_heat the oven to reach this temperature and turn it off, or leave the same temperature for all the rising time.
You will note with pleasure that your dough will rise a lot: I have a KitchenAid and usually the dough raises over the edges of the bowl: such a satisfaction!
Now, it is time to knead the dough: be careful.
Gently remove the cling film: part of the dough will have stuck to it. Wet your hand to favor the operation or use a wet spoon. The dough will deflate, that’s ok.
Butter aluminum molds or use the silicon ones, which I prefer definitely, you don’t have neither to grease nor to flour them.
Be very patience because the dough is sticky and elastic (this is the result of a strong flour combined with 5 eggs): you need a little skill;
Wet your hands and take a piece of dough enough to fill 1/3 of the mold; cut it with your fingers, a sudden pinch is ok, don’t use a knife. Drop it into the mold;
Cover each mold and allow the dough to rise until it has just started to rise above the edges of the molds: the dough is very strong, so it may take little time to;
Pre-heat the static oven at 180°C (356°F):
Bake babà for 20-30 minutes; if they get too brown at there end of the baking time, decrease the temperature to 160°C (320°F). During the baking process they will rise, that’s good, well done! You babà will be perfect;
Let them col down completely in their molds;
In the meanwhile prepare the stuffing, let’s call it syrup;
Bring 500 ml of water with the sugar too a rolling boil; the sugar must be completely dissolved;
Add the rum or the liquor to your taste;
Add the zest of an organic lemon;
Remove from the heat and let cool down to 35°C (95°F);
Transfer the syrup to a large bowl;
Place the babà in the bowl and soak them in the syrup for about 10 minutes; turn them several times with a perforated spoon to soak up the syrup;
As babà are very spongy, you must squeeze them;
Delicately squeeze them with your finger as you were squeezing a sponge; in this way the syrup will get into the babà as well;
Let them dry up on a rack, on a plate or in a container to save more syrup;
If you want to make your babà even more tempting, polish them. Brush the baba with 50g of apricot jam diluted with 1 or 2 spoon of rum syrup; they will eat glossy and very mouthwatering;
You may half babà and stuff them with whipped cream or custard.
Serve and enjoy them immediately.