FOR THE DOUGH :

1. In the bowl of a stand mixer, dissolve crumbled fresh yeast in water and lukewarm milk with a spoon. Let sit for about 15 minutes.
2. Pour in sugar, eggs, rum, vanilla seeds, flour and salt, and mix on low speed with a paddle attachment for about 10 minutes.
3. Add butter softened at room temperature in a few additions, and knead on low-medium speed until the dough becomes smooth and doesn’t stick on the sides anymore, about 30-40 minutes.
4. Cover the bowl with cling film and let rise at room temperature until the dough doubles in size, about 1 hour.
5. Re-shape the ball by hand a little bit to release the air from the dough. Put back in the bowl, cover with cling film, and place in the refrigerator for 4 to 12 hours (ideally overnight).

SHAPE THE DOUGH:

1. Remove the dough from the refrigerator and reshape a little bit. Grease a tinned-steel brioche mold.
2. Shape into a thick oblong log and cut a quarter (1/4th) of the dough, that you will use for the brioche head.
3. Shape the bigger part (the remaining 3/4th of the dough) into a smooth ball, hiding the folds underneath. Using 3 fingers joined together, pierce the dough, making a hole large enough so it looks like a donut. Carefully transfer the dough into the brioche mold.
4. Take the remaining quarter (1/4th) and shape into a small ball. Using your hands, carefully roll out one side so it looks like a big drop. Place the thiner part of the head ball into the donut hole, and press with floured finger tips all around to make sure it seals the sides.
5. Brush with the egg mixture, and keep the excess for later.
6. Loosely cover the brioche with a cloth to prevent the dough from hardening, and let rise in a warm environment (from 77-86 F / 25-30 C) for about 1 hour and 30 minutes. The brioche must double in size.
7. Preheat the oven to 350°F (180°C).
8. Brush one more time with the egg mixture and bake for 20-30 minutes, until the brioche is golden brown. Transfer to a wire rack and let cool completely.