Lightly butter your chosen moulds and set them aside. In a mixing bowl, combine the yeast mixture with the flour, salt, softened butter, and honey.
With the mixer on low speed, add the eggs one at a time, mixing well after each addition until you have a smooth, elastic batter. Transfer the batter to a piping bag and fill each mould halfway. Cover loosely and let rise in a warm, draft-free spot for about 30–45 minutes, until the batter has visibly increased in volume.
Meanwhile, split the vanilla pods lengthwise and scrape out the seeds. Place both seeds and pods in a saucepan with the water, sugar, citrus zest, juice, and rum. Bring to a gentle boil, stirring until the sugar dissolves, then remove from the heat to let the flavours infuse.
Preheat the oven to 180 °C (356 °F) and bake the babas for around 30 minutes, until golden and cooked through. While still warm, remove them from their moulds and submerge completely in the warm syrup, letting them soak until fully saturated.
Serve the babas warm with a scoop of vanilla ice cream or a generous spoonful of softly whipped cream.
Enjoy the baba au rum!
