In the bowl of a stand mixer, dissolve the yeast and sugar in warm water. Stir to combine and allow to sit for 5 to 7 minutes. The mixture will bubble and smell like beer once the yeast as activated. Whisk in eggs until thoroughly combined.
Attach dough hook to stand mixer. Add salt and 2 cups of flour. Start the mixer on low speed until the dough begins to come together, about 2 minutes.
The dough should look like a shaggy mess. Increase speed to medium and add the softened butter one tablespoon at a time. The mixture will be very sticky. Add the remaining 2 tablespoons of flour. Continue mixing for another 3 minutes until the dough is shiny and smooth.
Stop the mixer and cover the bowl with plastic wrap or a kitchen towel. Allow the dough to sit for 15 minutes. Mix on medium speed for another 5 minutes. At this point, the dough will still by quite sticky. Do not add additional flour. Use a spatula or dough scraper to remove the sticky dough from the mixing bowl. Transfer the dough to a well-greased large bowl. Cover with plastic wrap or a kitchen towel. Allow the dough to rise at room temperature for one hour.
Punch down the dough. Fold the dough into thirds like a business letter. Place the dough back into the greased bowl. Cover with plastic wrap and allow to chill in the fridge overnight for at least 6 hours, and up to 12 hours.
SHAPE THE BRIOCHE LOAF:
Grease a 9x5-inch loaf pan. Punch down the chilled dough. Divide into 5 equal portions. Working with one portion at a time, roll the dough into a 5-inch square. Using the heels of your palms, gently flatten and stretch the dough to the desired size. Roll the dough into a log and place it seam-side-down in the loaf pan. Repeat with the remaining portions of dough.
Loosely cover the dough in the pan with plastic wrap or a kitchen towel and allow the dough to rest at room temperature. Leave the dough undisturbed until the dough reaches the top of the loaf pan. This will take anywhere from 90 minutes to 2 hours.
ADD THE EGG WASH:
Preheat the oven to 350 degrees F. Mix the egg wash by whisking together the egg, salt, and water until smooth. Remove the plastic wrap and lightly brush the egg wash over the top of the bread loaf.
BAKE THE BREAD:
Bake the bread for 35 to 40 minutes, rotating midway through baking. The finished bread loaf should be golden brown on top and the internal temperature of the bread should register at least 200 degrees F on an instant-read kitchen thermometer. Allow the bread to cool in the pan for 10 minutes. Remove the bread from the pan and allow it to cool to room temperature on a wire rack. Once the bread has cooled, use a serrated knife to slice the bread. Store the cooled bread in an airtight container at room temperature for up to 4 days.
Notes
This dough is best mixed using a stand mixer. This enriched dough requires heavy mixing and kneading, if done by hand.
Overnight chilling aids in the flavor development of dough.