In the bowl of a stand mixer, add the flour, sugar, yeast and salt. Lightly whisk together.
In a microwave safe measuring cup, add the buttermilk and microwave it in 15 second increments until it registers 90° on an instant read thermometer, about 30 seconds.
Add the egg and the buttermilk to the bowl with the flour mixture. Using a wooden spoon, loosely mix together the dry and wet ingredients just until they start to combine. Place the bowl back on the stand mixer, attach the dough hook and mix on medium-low (number 2 on a Kitchen Aid) until a soft dough starts to form, 2-3 minutes.
Scrape down the bowl and mix on medium-high speed (number 6 on a Kitchen Aid) for another 5 minutes. The dough should start to become very smooth and elastic.
Reduce the speed to medium-low (number 2 on a Kitchen Aid) add a few cubes of the softened butter and mix until it is fully incorporated into the dough, meaning that there is no buttery residue left around the sides of the bowl. Add a few more cubes, and again, mix until it is fully incorporated into the dough. Do this until all of the butter has been added. Remove the dough from the bowl and knead by hand for a couple of minutes on the counter. Move the dough to a large, lightly-oiled bowl, cover with cling wrap and a kitchen towel, and let rise until doubled, approximately 1 hour.
After the dough has doubled in size, remove it from the bowl and dump it onto a clean work surface. You can lightly flour the surface if needed. This dough is soft, but easy to work with so I try not to use any additional flour.
Line a large sheet pan with parchment paper and set aside.
Using a digital scale, divide the dough into 8 equal portions weighing approximately 100 grams each. Cover the portioned dough with cling wrap or a clean kitchen towel so it doesn't dry out while shaping the buns.
Shape each portion of dough into a smooth round ball by gently kneading it together, then using one hand to pull the sides taut around the ball while simultaneously using the fingers of the opposite hand to gently push the bottom of the dough in an upward motion. Pinch the bottom of the dough together. You should have some semblance of a ball. Place the ball on the counter with the pinched seam on the bottom, cup your hand loosely around the ball and roll it around gently on the counter in a small circular motion. You should be able to feel the ball of dough loosely moving around within your cupped hand. The tension of the dough against the counter will close the pinched seam on the bottom of the ball and also form a taut gluten skin. Repeat this action with each ball of dough
Add the balls of shaped dough (seam down) onto the parchment lined sheet pan in the configuration shown above. Cover with a kitchen towel and let them rest for 10 minutes. Then take a smooth, flat bowl scraper or a small flat bottomed metal bowl and press down firmly on each ball of dough. Gently peel the scraper (or bowl) from the dough. The balls should flatten into smooth round discs approximately 4" in diameter. You can also press them out using your fingers. If the dough seems tight and tries to spring back too much, cover them with a kitchen towel and let them rest an additional 10-minutes. Then try to flatten them out again. There will be a little bit of spring back, but they should mostly hold their shape and size.
While the shaped buns are rising on the counter, place a rack toward the center of the oven and preheat the oven to 375°
Lightly mist the shaped buns with water and cover with a clean kitchen towel. Let rise for 45 minutes to 1 hour. The buns should double in size becoming very puffy.
Once the buns are puffy and ready to bake, prepare the egg wash. Whisk the egg with the salt and lightly brush the buns with the egg wash. Bake 20-25 minutes. The tops should be a deep golden brown and the internal temp should register 190° on an instant read thermometer.
Remove the buns from the oven and move them from the pan to a wire rack to cool. Let buns cool completely before serving. *See note 2 below.