In a medium bowl, combine 1/4 cup (60 ml) of the milk with the yeast. Let rest for 5 minutes.
Add the remaining milk, egg, egg yolk, flour, sugar and salt. Mix vigorously with a wooden spoon until smooth. Stir in the butter. Cover with plastic wrap and let rise for 45 minutes at room temperature, or until the dough has doubled in volume.
With the rack in the middle position, preheat the oven to 400°F (200°C). Butter 24 mini muffin cups.
Punch down the dough to deflate. Using a 1-tbsp (15 ml) ice cream scoop, place a ball of dough into each muffin cup. Let rise for 15 minutes or until the dough has doubled in volume.
Bake until the babas are nicely golden, about 15 minutes. Let cool in the muffin cups. Unmould and transfer the babas to a 13 x 9-inch (33 x 23 cm) baking dish.
In the meantime, in a pot, bring the sugar and water to a boil, stirring until the sugar has dissolved. Remove from the heat and add the rum. Pour the hot syrup over the babas while they’re still hot and let soak for 20 minutes, gently turning them over after 10 minutes. The babas will keep in an airtight container for 3 days at room temperature. Heat them in the microwave for a few seconds before serving, if desired.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.