Begin by dissolving the yeast in warm milk in a small bowl. Let it sit for about 5 minutes until it begins to foam.
In a large mixing bowl, combine the flour, sugar, and salt together.
Make a well in the center of the flour and add the yeast mixture, eggs, and softened butter.
Mix everything together until a smooth, elastic dough forms. This can be mixed by hand or with a stand mixer fitted with a dough hook attachment.
Cover the dough with a clean kitchen towel and let it rise in a warm place for about 1-2 hours, or until doubled in size.
Preheat your oven to 180°C (350°F).
Grease baba molds with butter.
Divide the dough into equal portions and shape into balls, place the dough balls into the greased baba molds.
Bake the babas in the preheated oven for 20-25 minutes, or until they are golden brown and cooked through.
While the dough is baking, prepare the rum syrup. In a saucepan, combine water and sugar. Bring to a boil, then reduce the heat and simmer for 5 minutes. Remove the syrup from the heat and carefully stir in the dark rum.
Once the babas are baked, remove them from the oven and immediately pour the rum syrup over them, allowing it to soak in.
Leave them cool slightly in the molds before carefully removing the babas and placing them on a serving platter.
Serve the Baba au Rhum with whipped cream and fresh berries on the side.