In a medium sized bowl, add the warm milk, sugar, and yeast. Let bloom for 5 to 10-minutes until frothy.
As the yeast mixture blooms, combine the bread flour, mashed potato, and salt in the bowl of a stand mixer. Stir to combine.
Add the yeast mixture to the flour mixture and knead on low speed for 2 to 3-minutes until a shaggy dough forms, scraping down the sides of the bowl with a spatula as needed.
Add the eggs, and continue to mix for 2 - 3 minutes.
Increase the speed to medium low, add in the softened butter. Knead for an additional 5 minutes until the butter is fully incorporated and the dough starts to smooth out.
Turn the mixer off and let the dough rest for 10 minutes.
Knead the dough on medium-low speed for an additional 3 minutes until soft.
Using floured hands (the dough will be quite sticky) turn the dough out onto a lightly floured work surface, and form into a tight ball. (Process pictures are provided in the blog post above.)
Transfer the dough to a large bowl and cover. Let rest for 1-hour, or until doubled in sized.
Turn the dough out onto a lightly floured work surface, and punch the dough down. Divide into 12 equal pieces (approximately 75g each) and roll into individual dough balls.
Place the dough balls onto parchment paper lined baking sheets. Cover with damp kitchen towels, or lightly greased plastic wrap and allow to proof for an additional hour.
Towards the end of the second proof, preheat the oven to 400°F.
Mix the egg white and water in a small bowl and gently brush over buns.
Bake for 15 to 17-minutes until golden brown. Remove from oven and transfer to a wire rack to cool completely before slicing.
Notes
These are best when halved and toasted in a pan with butter.