In the bowl of a stand mixer, dissolve 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 has activated. Whisk in eggs until thoroughly combined.
Attach dough hook to stand mixer. Add salt and 2 cups flour to mixing bowl. 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 softened butter one tablespoon at a time. Mixture will be very sticky. Add remaining 2 tablespoons of flour. Continue to mix for another 3 minutes until dough is shiny and smooth.
Stop mixer and cover bowl with plastic wrap or kitchen towel. Allow dough to sit for 15 minutes. Mix on medium speed for another 5 minutes. At this point, the dough will be quite sticky. Do not add additional flour. Use a spatula or dough scraper to remove sticky dough from mixing bowl. Transfer dough to a well greased large bowl. Cover with plastic wrap or kitchen towel. Allow dough to rise at room temperature for one hour.
Punch down dough. Fold into thirds like a business letter. Place back in greased bowl. Cover with plastic wrap and allow to chill in fridge overnight for at least 6 hours, up to 12 hours.
Grease a 9x5-inch loaf pan. Punch dough chilled dough. Divide dough into 5 equal portions. Working with one portion at a time, roll dough into a 5-inch square. Using the heels of your palms, gently flatten and stretch the dough to the desired size. Roll dough into a log and place seam side down in prepared loaf pan. Repeat with remaining portions of dough.
Loosely cover dough with plastic wrap or kitchen towel and allow dough to rest at room temperature. Leave dough undisturbed until dough reaches the top of loaf pan. This will take anywhere from 90 minutes to 2 hours depending on room temperature and humidity.
Preheat oven to 350 degrees F. Mix egg wash by whisking together egg, salt, and water until smooth. Remove plastic wrap and lightly brush egg wash over the top of bread loaf. Bake bread for 35 to 45 minutes, rotating pan 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 bread to cool in pan for 10 minutes. Remove bread from pan and allow to cool to room temperature on wire rack. Once bread has cooled, use a serrated knife to slice. Store cooled bread in an airtight container at room temperature for up to 4 days.