In a small bowl, combine the warm milk and 1 tablespoon of sugar. Sprinkle the active dry yeast over the top and let it sit for about 5 to 10 minutes, or until it becomes frothy.
In a large mixing bowl, whisk together the eggs, melted butter, and the remaining sugar until well combined. Add the frothy yeast mixture and mix until smooth.
In another bowl, whisk together the flour and salt. Gradually add the dry ingredients to the wet mixture, stirring with a wooden spoon or spatula until a rough dough forms.
Transfer the dough to a lightly floured surface and knead for 8 to 10 minutes, until it becomes smooth and elastic.
Form the dough into a ball and place it in a greased bowl, turning it to coat all sides with butter. Cover with a kitchen towel and let it rise in a warm place for about 1 to 2 hours, or until doubled in size.
Once the dough has risen, punch it down gently to release the air. Transfer it to a floured surface and shape it into a loaf.
Cover the shaped dough with the kitchen towel again and let it rise for another 30 to 60 minutes, until it has puffed up nicely.
Preheat your oven to 350°F (175°C) while the dough is rising.
In a small bowl, whisk together 1 egg and 1 teaspoon of water. Brush this mixture over the top of the loaf for a beautiful golden crust.
Bake the brioche in the preheated oven for 25 to 30 minutes, or until the top is golden brown and the loaf sounds hollow when tapped on the bottom.
Remove the brioche from the oven and let it cool in the pan for about 10 minutes. Then, transfer it to a wire rack to cool completely.
