Combine the warm water, sugar, and yeast in a large bowl. Whisk to dissolve the yeast. Leave for 5-10 minutes until foamy.
Add the flour, salt, eggs, and rum. Knead well using the attachment on your mixer or by hand.
Slowly incorporate the butter cubes and knead well. The dough has to be glossy and smooth.
Cover the bowl with plastic wrap and let rise in a warm, draft-free place until almost doubled in size, about two hours.
Punch the dough down. Put in the fridge to chill for 12-16 hours or overnight. (The dough will be more delicious if you leave it to chill overnight. If you don't have the time, leave that part out.)
Cut the dough into 70g pieces. Shape each piece into a smooth bun and place on a baking sheet lined with parchment paper and dusted with flour. Leave plenty of room between the donuts to proof. Cover lightly with plastic wrap and leave to proof for about 1 hour, or until about doubled in size.
Heat 2 inches of oil in a heavy pot (cast iron) to 330-350°F. To fry donuts, drop several of them at a time and allow them to become golden before turning to the other side, about 1-2 minutes per side. Remove donuts from the oil with the spider tool and place them on a tray lined with paper towels.
Once slightly cooled, place 1/2 cup granulated sugar in a dish, roll each donut in sugar, and set aside to cool completely before filling.
For the vanilla cream, add the sugar, cornstarch, and milk to a large saucepan and mix until smooth. Cook, stirring continuously, over medium heat until mixture begins to thicken and boil. Reduce heat to medium and simmer for 1 minute. Remove from heat.
Add a little bit of mixture to the egg yolks and whisk together, then add egg mixture to milk mixture. Place the saucepan back on the heat and bring to a light boil. Allow boiling for 2 minutes, stirring continuously.
Remove from heat, add butter and vanilla. Stir until smooth. Pour into a medium bowl. Cover with plastic wrap and press it directly on the surface. Refrigerate until cold.
Fill a piping bag with a round tip with the vanilla cream. Using a small knife, cut a hole in the side of each donut about 3/4 of the way into the donut and pipe a generous amount of vanilla cream into the center of the donut.