Place the 1/4 cup of warm water in a cup; stir in the 1 teaspoon of granulated sugar. Sprinkle the yeast on top and let stand until foaming, about 5 minutes.
Place the 1 1/2 cups of flour and a pinch of salt in the bowl of a mixer and make a well in the center. Pour in the yeast mixture and add 2 of the eggs. Blend on medium speed until smooth and elastic, 5 minutes. Cover the bowl with plastic wrap and let stand in a warm place until the dough has doubled in volume, about 1 hour.
Preheat the oven to 375 degrees . Butter a 9 1/2-inch (6-cup) metal ring mold, then dust it with flour and set aside. Add another egg to the dough and beat until incorporated. Mix in the last egg and continue to beat until smooth, about 5 minutes. Scrape the dough evenly into the prepared pan. Cover loosely with plastic wrap and let rise in a warm place until the batter is 1/4 inch below the top of the pan, about 45 minutes.
Remove the plastic wrap and bake the cake until puffed and golden, 15 to 18 minutes. Cool on a rack for 5 minutes, then loosen the edges with a thin, sharp knife and invert onto the rack. Set the rack over a plate to catch the rum sauce.
In a small saucepan, bring 1 cup of water and the 1 cup of granulated sugar to a boil. Remove from the heat and stir in 1 more cup of water and 1/2 cup of the rum. Gradually spoon the syrup over the warm cake. Transfer the cake to a serving plate and chill for 1 hour.
Before serving, make the glaze: In a small bowl, stir together the confectioners' sugar, the remaining 1 tablespoon of rum and 1/8 teaspoon of salt. Drizzle over the chilled cake and serve.