Combine water and milk in a small sauce pan. Heat over medium high heat until an instant read thermometer reads 90°F to 100°F.
Add active dry yeast and sugar, stirring to combine. Allow to bloom 5 to 10-minutes until active and frothy.
While yeast mixture rests, add salt and bread flour in the bowl of a stand mixer, whisking to combine.
In a small bowl, beat egg and egg yolk together.
Add the yeast mixture and beaten eggs to the dry ingredients. Turn mixer on low speed until combined, scraping down the sides of the bowl as needed.
Once you have a rough dough, add the unsalted butter and increase the speed to medium-low. Knead for 10-minutes until butter is combined and dough begins to pull away from the sides of the bowl and becomes smooth. (The dough will be slightly sticky at this stage).
Turn the dough onto an un-floured work surface, and shape into a ball.
Place dough into a large mixing bowl and cover in an airtight container, or a damp kitchen cloth. Let rest for 1-hour for the first proof.
After the first proof, punch the dough down and turn it onto an un-floured work surface.
Divide into 8 equal sized pieces (approximately 80 grams each). Shape into individual balls by folding the edges of the dough onto themselves, and then scooping and dragging across the work surface.
Place the balls on a parchment paper lined baking sheet, and flatten into discs that are approximately 3 to 4-inches wide. Cover with a damp kitchen towel and let proof for 2 additional hours.
Towards the end of the second proof, preheat oven to 375°F and combine the egg wash ingredients in a small bowl.
Uncover the dough, and brush evenly with the egg wash.
Bake for 15 to 20-minutes until golden brown.
Remove from the oven and allow to cool on the tray for 5 to 10-minutes. Transfer to a wire cooling rack and allow to completely cool for approximately 1-hour.
Split with a bread knife and optionally toast with butter until golden brown.
Notes
Here are some of the nitty gritty specs for all my bread nerds out there (don't worry - you're in good company): This recipe yields one 640g Dough Ball at approximately 65% hydration. This will make 8-buns at 80g each which are about the perfect serving size. They will seem small before the second proof, but they almost double in size by the time they are done baking.You can also scale this recipe up or down, just be sure to use baker's percentages and weighing your ingredients and not volume measurements.