Pour the flour into the bowl of a stand mixer fitted with a paddle attachment. Add the yeast and mix well. Add the eggs, one at a time, mixing between additions. When they are blended in, add the sugar and mix again. Add the salt and let it continue mixing for a few more minutes overall it will take 10 minutes.

Add the butter pieces, mixing between additions. Let each piece be worked into the dough completely before adding the next one.

Once the butter is combined, switch to the hook attachment and continue to mix for 10-15 minutes. 

Transfer the dough to another large bowl. Cover the bowl with plastic wrap and leave to rise in a warm place for about 3 hours.

When you are just about ready to bake, grease the molds with butter.

Remove the plastic wrap from the dough. With a wet hand take a portion of the dough (about 2 oz), forming a small ball and dropping each piece into one of the buttered molds.

Cover molds with an apron or towel without any fluff and leave to rest until the dough rises over the edge. This will take about 45 minutes. Alternatively, you can spray the underside of a piece of saran wrap with a bit of PAM and lay that over the dough as it rises, sprayed side down. 

Preheat your oven to 360F. Move the babà molds onto a baking sheet, spreading them out so that they aren't touching each other. Bake for about 20-25 minutes, until the babas have risen and are golden.

Let them cool for about ten minutes, remove from them molds and place them on a wire rack to finish cooling.

Pour the water in a saucepan on medium-low heat.

Add the sugar, stir until completely dissolved.

Turn off the heat, add the rum and wait for the water to reach 120°F (50°C), the ideal temperature for soaking. You can measure the temperature with a candy thermometer. 

Dip in 1 or 2 babàs into the syrup. Let them soak for about 1 minute turning them over. Give them a gentle squeeze 1 or 2 times: this will soften them and improve the soaking.

After the last dip, if you prefer, can stop squeezing them. Drain them on a wire rack, for a couple minutes.

Serve them plain or garnished with whipped cream and berries.