This spiced version of baba au rhum is so festive for the holidays.
Sift flour and salt into large bowl. In smaller bowl, combine warmed milk (110 degrees), 4 teaspoons granulated sugar, and yeast. Let sit 5 minutes until foamy; stir. With wooden spoon, combine yeast mixture into flour. Add 1 egg, lightly beaten, and butter. Stir 5 minutes; dough will still be sticky. Cover; set in warm place 45 minutes, until doubled.
Meanwhile, heat oven to 425 degrees. Beat dough with wooden spoon to remove air bubbles, adding 1 to 2 tablespoons flour if needed. Turn out dough onto lightly floured board. Form into 1-inch balls with lightly floured hands. Place dough in well-buttered 1-inch-by-2-inch molds. Cover, and let proof 10 minutes. Whisk remaining egg with 1 tablespoon water. Brush the baba au rhum with the egg glaze, and sprinkle with sugar crystals. Bake about 10 minutes, until golden.
Meanwhile combine remaining 1 cup granulated sugar, 3/4 cup water, rum, and ginger in small saucepan. Stir over medium heat until sugar dissolves, about 3 minutes. Cook 5 minutes more. Set aside to steep. When baba are golden brown, remove from oven. Strain syrup into a large bowl. Turn baba out of pan and into syrup to soak. May be made up to this point 3 to 4 hours ahead. Serve baba in syrup at room temperature.