Heat the water in a bowl until it's warm but not too hot. You should be able to dip a finger in comfortably. Stir in the yeast and sugar till dissolved. Cover and set aside until foamy, about 10 minutes.
In the bowl of a stand-mixer fitted with the dough hook, (or a large mixing bowl if doing this by hand), combine the yeast mixture, egg, rum and vanilla. Mix and then add the flour and salt. Knead on low speed till a shaggy dough begins to form.
Add the butter, one cube at a time and mix on medium speed, adding the next cube after the first is mostly incorporated. I accidentally added them all at once, so had to work a little harder to get the glossiness, so gradually is best.
Knead for 6 to 8 minutes in the mixer (about 10 minutes by hand) until the dough is smooth and glossy. Cover tightly with clingfilm and set aside for 1 to 2 hours until doubled. Even in a hot kitchen, this dough will rise slower because of the butter and eggs.
If the risen dough feels sticky, place it in the fridge for 15 minutes to chill and become easier to handle.
Form the dough into 5 large or 10 small rounds (see notes). Place at least 2 inches apart on a silicone or parchment lined sheet. Cover loosely with clingfilm and allow to double, about 1 more hour.I found them becoming wider rather than taller but the final texture remained unaffected.
When the dough balls have almost doubled, pour vegetable oil about 2/3rds of the way in a large, sturdy wok or pot. Pinch off a small bit of the dough and drop it into the oil. If it rises to the surface and begins to brown, the oil is hot enough.
Carefully lift off the puffy dough balls, taking care not to deflate them as much as possible. If some lose their shape or become wrinkly, do not fret. They will still fry up well. Just slide them into the hot oil and cook for two to three minutes on each side until golden-brown. Place on a paper-towel to drain.
Allow the fried donuts to cool completely. Meanwhile, peel the mango and place the pulp in a blender. Puree until completely smooth.
To make the filling, place a mixing bowl and beaters from a hand mixer in the fridge about 20 minutes before beginning. This ensures that the cream stays cold while you whip it.
Pour the cream into the bowl and whip on medium speed with a hand mixer until soft peaks form and hold their shape when you take the beaters out.
Fold in the mango puree slowly. The cream will deflate and thin out a bit because of the water that mangoes contain, don't worry about it. Place the cream in the fridge while you prep the donuts.
Use the end of a fork to poke a hole in the side of each donut, taking care not to poke all the way to the end. Move the fork around to loosen the donut just slightly.
Fill a piping bag or bottle with the filling, and attach a thin tip which won't break the donut. Fill each one till the mango cream reaches the outside, then stop. Dust liberally with powdered sugar and serve immediately.
The donuts are naturally best eaten fresh, but keep well in the fridge for 2 to 3 days. Preferably fill them just before you eat. They remain soft enough to be eaten cold! Enjoy :)