In the bowl of a stand mixer fitted with the dough hook, combine the flour, sugar, salt, and yeast. Mix on low speed for about 30 seconds until just combined.
In a separate bowl, whisk together the 4 large eggs. With the mixer on low speed, gradually add the whisked eggs to the dry ingredients. Mix until a shaggy dough forms.
Increase the mixer speed to medium-low and knead for about 8-10 minutes, or until the dough is smooth and elastic. It will be a bit sticky.
With the mixer running on low speed, gradually add the softened butter, one tablespoon at a time, allowing each piece to incorporate before adding the next. Continue to mix until all the butter is incorporated and the dough is smooth and pulls away from the sides of the bowl.
Transfer the dough to a lightly oiled bowl, cover with plastic wrap, and let it rise in a warm place for 1-1.5 hours, or until doubled in size.
Punch down the dough and turn it out onto a lightly floured surface. Shape the dough into a loaf and place it in a greased 9x5 inch loaf pan. Cover loosely with plastic wrap and let it rise again for about 45-60 minutes, or until it has almost doubled in size.
Preheat your oven to 375°F (190°C). In a small bowl, whisk together the beaten egg and water for the egg wash.
Gently brush the top of the risen dough with the egg wash.
Bake for 30-35 minutes, or until the crust is golden brown and the internal temperature of the loaf reaches 190°F (88°C). If the top starts to brown too quickly, loosely tent it with foil.
Let the brioche cool in the pan for 10 minutes before removing it to a wire rack to cool completely.
Notes
For a richer flavor, you can let the dough undergo a cold fermentation in the refrigerator overnight after the first rise.