For the baba: Mix together the flour, butter, honey, yeast and salt in a large bowl. Knead the dough while adding the eggs one by one. The dough must not stick to the sides of the bowl; knead until the dough becomes elastic and smooth, about 45 minutes. Leave to rest for 30 minutes in the bowl.
Preheat the oven to 350 degrees F/180 C. Grease 18 molds with butter.
Distribute the dough evenly into the molds by hand or using a pastry pipe, filling halfway. Place into a deep baking tray, and fill with warm water (85 to 95 degrees F) up to the top of the molds. Let rest until the dough rises and overflows the molds slightly.
Bake for about 25 minutes, turning the tray half way through. Cooking time varies according to size, color and dryness. Unmold the babas, and let dry for about 6 minutes in an open door oven, and then cool on a rack.
For the syrup: Mix 2 1/8 cups water with the sugar, zests and vanilla in a saucepan and bring to a boil. Remove from the heat and let the flavors infuse. Dip the babas into the hot but not boiling syrup, turning with a slotted spoon. Check the dipping. The babas must be imbibed and properly drained.
For the cream: Mix the vanilla seeds into the cream. Whisk in the sugar so to obtain a three-quarters whipped cream (must remain slightly "wet").
Place the babas on a plate and coat with the apricot topping. In front of the guests, cut the babas in two and cover with the aged rum of their choice. Serve with the vanilla cream.