When I eat a good baba au rhum (or au some other booze), I'm convinced that it's the best dessert ever. (There are lots of things like that - trifle, floating island, anything with caramel sauce - so don't take the superlative too seriously.) The trick is to find a baba so wonderful that it triggers such a reaction. My touchstone is Alain Ducasse's famous version at his restaurant in Monte Carlo, for which he charges €30 ($33); this seems excessive until you see it come to the table in a special silver dish with a hinged dome for a lid, accompanied by a whole cart of rums from which to choose. Another great fancy-restaurant baba is served at Hélène Darroze at The Connaught in London, where it is doused not with rum but with excellent Armagnac bottled by Ms. Darroze's brother; when Jackie and I last ate one, it was cleverly rendered seasonal by the chestnut puree folded into the accompanying whipped cream.
It's not a thing I regularly make at home, though I've done it often enough that the basic recipe is included on the sheet clipped to the refrigerator door (along with a fine chocolate pudding and Jackie's Polish apple pie). Because it is so soft and (initially) sticky, the baba dough I like can be scary - and it teaches a useful lesson in not being frightened of recipes that look as if they've gone wrong but haven't.
These quantities make six gigantic babas, each big enough for two portions: a large half-baba, cut side up, looks great - and since no one will see the crust you don't need to glaze it with melted, strained jam as you'd have to do if you worked in a pastry shop or restaurant.
I started the day before. In a powerful food processor, I pulsed to combine 9 ounces (250 g or two cups) flour, 1 generous teaspoon instant yeast, 1/2 teaspoon fine salt and 2 teaspoons sugar. I added three eggs and 3 fl oz (90 ml or 6 tablespoons) whole milk, all at room temperature, and turned on the machine. When this came together, it was very soft and sticky, and this would only get worse: I next thoroughly beat in 3 oz (85 g or 6 tablespoons) unsalted butter, also at room temperature but not anywhere near melting. I let the machine run until the dough was homogeneous, scraping down the sides once or twice: the butter has to be fully incorporated. Now I had a hell of a mess which I needed a rubber spatula to scrape into a container with a lid; at this stage there was no point in attempting to form it. (Think of Jim Lahey's famous no-knead bread, popularized by Mark Bittman: this is just as umpromising-looking a mass of dough.)
I covered the container and left my mess at cool room temperature for half an hour or 45 minutes. Amazingly, it had begun to rise, and even more amazingly, now that the flour was fully hydrated and other wonders of cooking science had been at work, it was possible to pick up the dough and form it into a ball. I put the lid onto the container and let the dough rise to at least double its original size, then deflated it, re-formed the ball and put the container into the fridge overnight. In the refrigerator it continued to rise for a time, but slowly.
Now I made the syrup in which the babas would be soaked. The liquid in a baba is not mainly rum (or other spirits); it is mostly sugar and water with flavorings. In a saucepan I combined 2 cups (500 g) sugar with 4 cups (1 liter) water and brought this to the boil to dissolve the sugar. Off the heat I added a little bit of vanilla extract and the juice of half a big lemon (or a whole small one), and when it had cooled a while, half a cup (120 ml) of Barbadian dark rum (don't fret: there'll be more rum later). Use whatever rum you like the flavor of - or go the Darroze route and use Armagnac or Cognac. The vanilla and lemon will enhance the spirits' flavor; so will the sugar, come to think of it. Keep this syrup in a closed container; it doesn't need to be refrigerated, but if you do store it in the fridge be sure to let it come up to room temperature when you're about to bake.
Early the next day, I took the dough out of the fridge and heated the oven to 400 degrees F (a little over 200 C). I found my half dozen baba/dariole molds (2-5/8 in / 65 mm high and the same across at the top - narrower at the bottom) and did NOT butter them: Perhaps superstitiously, I believe that yeast doughs rise better when they have a non-greased surface to grab on to. The other side of that coin is that they can stick when baked and might need to be helped out of the molds with a narrow-bladed knife. You may find that buttering the molds (or using non-stick ones) is smarter, and you may be right.
I divided the still-cold dough into six equal portions (I used a scale to verify their weight) and formed each into a ball, being sure to stretch the dough surface to form a smooth top, with the seams below. I set the molds onto a baking sheet and covered them with a sheet of plastic wrap, then a kitchen towel. After about 90 minutes, when they had a bit more than doubled in volume and had fully come up to room temperature throughout, I baked them for 25 minutes (start checking after 20 - sooner if you have a convection oven). They rose enormously and beautifully and were a nice toasty brown on top. I poked one with a cake tester to make sure it felt well baked at the center: once it got through the crust, the probe met no palpable resistance.
When they were cool enough to handle, I got them out of their molds and left them on a wire rack to cool and to dry for a couple of hours before putting four of them in a close-fitting rectangular container and immersing them in the prepared syrup (all of it), using a weight to keep them submerged until they got syrup-logged and sank of their own accord. Be sure they are totally soaked, like a sponge, and keep them in the syrup. Nearer dinner time, I halved each baba lengthwise and continued to soak - and to baste whenever I thought of it.
To serve, I placed half-baba on soup plates and garnished them with little mounds of slightly sweetened and vanilla-ed whipped cream. At the table, I poured straight rum over each portion, enough to flavor the cakes and to spill over into the dish.
The babas themselves were light and just rich enough; the lemon and vanilla in the syrup added interest; the rum... well, it wouldn't have been a baba au rhum without it, would it?
The two unsoaked babas, Jackie and I treated like brioches (which they were, sort of): next morning we halved one, toasted it on a ridged grill pan and ate it with strawberry jam. Now, there's something you couldn't do with a €30 restaurant baba.