Dissolve yeast in cold water. In a stand-up mixer with paddle attachment place flour, sugar, salt and eggs. Mix 5 seconds to combine then add the water+yeast mixture and mix at medium high speed for 2 minutes.
Add cooled butter, keep mixing another 7 to 10 minutes.
Let rest, covered for 20 minutes, not more.
Pipe mixture in individual molds, ⅓ of the height of the mold.
Let rise, covered, in a warm spot until dough is about ¼ of an inch (6mm) from the top, then uncover and let rise to the top of the mold. The whole proofing can take up to 40 minutes.
Preheat the oven at 190C/375F and bake individual babas for about 15 minutes. A large savarin can take up to 40 minutes to bake. Unmold while still hot, place on a cooling rack for 30 minutes before soaking in syrup.
In a large pan add water, sugar, vanilla seeds and zests, bring to a boil then let cool for 30 minutes.
Place the babas on a cooling rack with a tray underneath and ladle the warm syrup on top or soak the babas for 3 minutes then let drain the excess syrup for 10 minutes. Ladle the dark rum and let drain the excess. Serve with whipped cream or with pastry cream and in the summer with slices of fresh fruit.
