Warm up 1/4 cup milk. But don’t make it too hot, think bathwater. Cut 3
Tbs. butter into pieces.
Preheat oven to 375 degrees F. Butter six ramekins. In the bowl of an electric mixer mix together (with a spoon) 1 envelope active dry yeast, 1 tbs. sugar and milk. Let it sit for about 5 minutes. If mixture doesn’t appear foamy, your milk might be too hot, or you might have bum yeast. Add 1/4 cup all-purpose flour, and stir until smooth. Cover bowl with a tea towel and let sit at room temperature for 30 minutes.
In a small bowl, whisk together 1 large egg and 1/4 tsp. salt. Add another 1/2 cup flour to the yeast mixture, alternating with the egg mixture. Beat well at low speed. Add the butter, one piece at a time. Divide the dough among the ramekins and let it sit in a warm place 30-40 minutes.
Bake 15 minutes or until golden.
Meanwhile, combine 1/4 cup apricot jam, 1/2 cup dark rum, and 1 Tbs. sugar in a saucepan. Bring to a boil, then reduce heat and simmer, stirring frequently, for 10 minutes.
Set cooked babas on a cooling rack and poke all over the top with a toothpick, then brush with a generous amount of syrup. Serve warm.