Heat half of the sugar with the milk just until lukewarm, then remove from the heat and sprinkle the yeast on top. Leave it for 10 minutes and the yeast should foam.
Place the yeast mixture into your Stand mixer with the kneading hook on, then mix it with the eggs. Mix in the dry ingredients: bread flour, the rest of the sugar, and a pinch of salt, and keep the mixer running to knead the dough until it comes together. This will take about 3 minutes.
Slowly add in the softened butter piece by piece while the mixer is on, and once all the butter is in, knead for around another 10 minutes on medium-high speed. The dough will become elastic and shiny. Once you have achieved the 'gluten window' (see photos above), stop kneading.
Move the dough into a bowl, cover it with plastic wrap, and let it rise at room temperature for 1-2 hours, or until the dough doubles in size. Then punch it out, re-shape it into a bowl, and put it in the fridge to rise again for a minimum of 8 hours or up to 24 hours.
The next day, divide the dough into 8 equal pieces and roll them into balls. Use a Silicone baking mat to help shape the dough without needing dusting flour. Grease a large (27x15cm / 11x6 inches) loaf pan tin with arrange the dough balls as 2-2-2-2.
Mix an egg with a splash of water, and brush an egg wash over the surface of the dough. Allow the brioche to proof for a further 1-2 hours until the dough doubles in size and fills the tin.
Preheat your oven to 180C / 356F towards the end of the proofing time.
Place the French brioche bread into the oven and bake for about 45 minutes until golden brown.
Remove from the oven and brush a small amount of melted butter over the top of the brioche. Leave it to cool for a few minutes before turning it out onto a wire cooling rack.