Step 1Dissolve the yeast in the warm water and whisk in the eggs.
Step 2Combine the flour, cocoa, sugar and salt in the bowl of a mixer. With the mixer on low speed, add the egg mixture to the dry ingredients. When the dough starts to look crumbly but has not yet come together, add the butter, a few pieces at a time. Increase the speed to medium and beat about half a minute until the butter is incorporated into the dough. This is a soft, sticky dough that will stick to the sides of the bowl.
Step 3Scrape the dough onto a lightly floured board and bring it together into a ball. Cover it loosely with plastic wrap, set it in a warm place and let it rise until doubled, 1 1/2 to 2 hours.
Step 4Grease 6 (6-ounce) ramekins with butter. Divide the dough into 6 pieces and roll each piece into a ball. Place each ball into a ramekin. Cover the ramekins with plastic wrap and let the dough rise about 45 minutes.
Step 5Heat the oven to 375 degrees. Bake the babas until the tops begin to darken, 12 to 15 minutes. Remove them from the ramekins and allow to cool completely.
Step 1Cook the sugar in the water in a small saucepan over medium heat until it dissolves. Bring the syrup to a simmer and remove from the heat. Stir in the rum.
Step 1Cut a 1-inch hole in the bottom of each baba and use your fingers to make a hollow in the center. Do this gently, or the babas will crack. While the syrup is still warm, dip the babas into the syrup, one at a time, turning each to saturate all sides and keeping each baba in the syrup for about 30 seconds. Place the babas on a rack and let them stand about 15 minutes to cool completely and absorb the syrup.
Step 2Whip the cream to firm peaks with the sugar and vanilla. Spoon or pipe the cream into the center of the babas. The babas can be served immediately or refrigerated up to several hours and moistened with additional syrup just before serving.