In a small bowl, dissolve the yeast in the 3 tablespoons of warm almond milk. Let it sit for about 15 minutes or until foamy.
To the bowl of a stand mixer fitted with the flat beater, add the flour, sugar, and baking powder. Mix for 5-10 seconds.
Next, add the diluted yeast, the rest of the almond milk, and the vanilla extract. Beat at medium speed for about 30 seconds. Add the butter and beat for another 2 minutes or until the dough is completely smooth. Note: the dough will be a bit runny and not as thick as a brioche dough, which is normal.
Transfer the dough to a glass bowl and cover it with plastic wrap. Let the dough rise for about 1 hour in a warm place (you can place the bowl in an oven with the light on). The dough should double in size.
Using a spatula, deflate the dough and transfer it to a piping bag.
Lightly grease a mini fluted tube pan with oil. Fill each cavity to the 3/4 with the dough using the piping bag. Let the dough rise for another 30 minutes. In the meantime, preheat the oven to 355 °F (175°C).
Bake for about 15 minutes, or until slightly golden brown. The cakes will be very soft out of the oven, so wait for at least 30 minutes before removing them from the pan.
Rum syrup
While the cakes are baking, combine the water, sugar, rum, vanilla extract, and orange zest in a medium saucepan.
Bring to a boil and let simmer covered for about 7-8 minutes. Set aside.
Dipping
Working with one at a time, dip each cake in the warm rum syrup and let it soak the syrup for about 2 minutes. Remove from the syrup using a spatula and place each cake on a rack to drain the excess syrup. Set the remaining rum syrup aside.
Refrigerate the rum baba for at least 3 hours. Just before serving, drizzle with more rum syrup and top with whipped coconut cream or custard, and fresh fruits. These rum babas taste better the same day but will keep for up to 4 days in the refrigerator.
Notes
Make sure all of your ingredients are at room temperature.
Check the expiry date of your yeast. If it has passed or is near that date, the yeast might not be active anymore.
Depending on the temperature of your home, the dough might take up to 1 and a half hours to rise. Adjust the rising time accordingly. The dough should double in volume after the first rise. If your home is too cold, place the dough in an oven with the light on.
Only fill the cavities of your pan to the 3/4. The dough will increase in volume after the second rise and during baking.
After a couple of days in the fridge, rum baba tends to become a bit drier. To keep them moist longer, store each baba in a small cup filled with 3-4 tablespoons of rum syrup and keep covered with plastic wrap. This way, the cakes will stay moist for up to 4 days.