**Activate the Yeast:**
In a small bowl, combine the warm milk, warm water, and sugar. Sprinkle the yeast on top and let it sit for 5-10 minutes, or until it becomes frothy.
**Prepare the Dough:**
In the bowl of a stand mixer, combine the flour and salt. Add the yeast mixture and eggs, and mix on low speed with a dough hook attachment until a dough forms.
**Knead the Dough:**
Increase the mixer speed to medium and knead the dough for 5-7 minutes, or until it becomes smooth and elastic. You may also knead by hand on a floured surface for the same amount of time.
**Incorporate the Butter:**
Gradually add the softened butter, one tablespoon at a time, and continue to knead until the butter is fully incorporated and the dough is smooth and shiny.
**First Rise:**
Place the dough in a lightly greased bowl and cover with a clean kitchen towel. Let it rise for 1-1.5 hours, or until it has doubled in size.
**Shape the Loaf:**
After the dough has risen, punch it down to release air. Turn the dough out onto a floured surface and shape it into a loaf by folding the edges in and rolling it into a tight cylinder.
**Second Rise:**
Place the shaped dough into a greased loaf pan. Cover the pan with a towel and let the dough rise for 30-45 minutes, or until it has doubled in size and slightly risen over the top of the pan.
**Preheat the Oven:**
While the dough is rising, preheat your oven to 350°F (175°C).
**Prepare the Egg Wash:**
In a small bowl, whisk the remaining egg. Brush the top of the dough with the egg wash for a golden finish.
**Bake the Brioche:**
Bake the brioche in the preheated oven for 25-30 minutes, or until the top is golden brown, and the loaf sounds hollow when tapped on the bottom.
**Cool and Serve:**
Remove the brioche from the oven and allow it to cool in the pan for 5 minutes before transferring it to a wire rack to cool completely. Slice and serve.