In your stand mixer bowl, add in warm milk, sugar and yeast. Let the yeast sit for 5 minutes to activate. While the yeast activates, mix 3 eggs using a fork. Now remove 2 tablespoon of the whisked egg into a separate bowl for egg washing. Add 2 tablespoon of milk into the 2 tablespoon of egg and mix well. Set aside for later.
Now the yeast must be activated. It should look foamy to look. Now add in the whisked 3 eggs, plain flour, salt into the stand mixer bow. Place the bowl in the stand mixer, use dough hook for mixing. Now start your mixer and start kneading the dough. Let the dough knead till the flour is hydrated with the milk and eggs.
At this point start adding in the soft butter little at a time while you keep on kneading. Let the dough knead on low speed till all the butter is incorporated. The dough will look like a shaggy dough once the butter is mixed in. Now increase the speed of the mixer slightly and knead the dough. Make sure to stop the mixer in between and scrape the bowl of the stand mixer. Continue kneading the dough for at least 15 to 20 minutes. By this time the dough will look glossy, smooth, shiny and soft.
Now let's check for window pane test. Window pane test is very important to check whether the gluten development in dough is formed. This helps in getting the texture of the bread. Take a small ball of the dough and stretch it into a square. Keep on stretching until it becomes a thin film in the middle without breaking. The dough must be thin enough to pass light through it. If it is obtained without any tear, then your dough is properly kneaded.
Remove the dough to a work surface and shape into a smooth ball like. The dough will be very easy to handle. Place the shaped dough back into the stand mixer bowl. Cover with a cloth and allow it to rest and rise for 1 to 2 hours till the dough is doubled in volume. Now after 1 to 2 hours the dough has risen more than double in volume.
Remove the dough on a work surface. Flatten it gently. Roll the dough into a log like. Divide the dough into three equal portions. Shape each portion into a smooth ball.
Take each portion of the dough, roll it into a rope like. Now bring the top ends together and stick it together. Start braiding the bread. Once you come to the end, tuck the end underneath and shape it into a smooth braid by rolling it back and forth.
Grease a 8x4 inch loaf pan generously with butter and set aside. Place the braided brioche loaf into the pan. Cover the loaf pan with a kitchen towel and allow it to rise for 45 minutes to 1 hour. Now the loaf has risen almost more than double in volume. Remember the egg wash which we prepared earlier. Brush the egg wash generously over the bread dough using a pastry brush.
During the last 10 minutes of proofing, make sure you preheat oven to 190 degree C | 375 Degree F. Once the oven is hot, place the loaf pan into the oven and bake for 35 minutes. If you find the brioche looking too dark half way through the baking, make sure to tent it with some aluminium foil loosely so the top doesn't burn before the bread is cooked inside.
Once the bread is baked, remove it from oven and allow it to cool in the loaf pan for 20 minutes before unmoulding from the pan. Now allow the brioche to cool completely before slicing.