In a stand mixer bowl fitted with a dough hook, add the milk and stir in the yeast and one tablespoon of sugar. Leave it to sit for 5-10 minutes until foamy. Add to it the remaining sugar, eggs, flour, and salt. Turn the mixer on low and combine until it forms a thick but slightly sticky dough. Mix this dough for around 5 minutes to begin developing the gluten.

Add in the butter, a cube or two at a time, ensuring each cube has been incorporated before the next addition. Turn the mixer on medium and keep it mixing until the sticky dough starts to strengthen and come together and pull away cleanly from the sides of the bowl.

For the best gluten development and brioche crumb, mix the dough for at least 15 minutes. Proper gluten development will allow you to stretch the dough so thin you can almost see through it. This is called the window pane effect.

Pull the dough from the bowl onto a bench and form it into a ball. Place the dough ball into a clean bowl, cover it with plastic wrap or a damp kitchen towel, and let it rise for 1.5-2 hours (depending on room temperature) until doubled in size. Deflate the dough gently and reshape it into a ball again. Cover it with a lid or plastic wrap and place it in the refrigerator overnight. This step can be shortened, but an 8-12 hour fridge proof gives the best flavor.

Let the shaped dough rise in a warm spot until it has doubled in size.

Preheat the oven to 375°F/190 °C and gently whisk an egg and one tablespoon of water in a small bowl. Brush the tops of the dough with the egg wash.

Bake the brioche bread for approximately 30 minutes until a deep golden brown.

Once baked through, it should sound hollow when tapped. If you have an instant-read thermometer, you can check for an internal temperature of 195°F / 90°C.