Place the flours, water, yeast, sugar, salt, and 5 eggs in the bowl of a stand mixer fitted with the dough hook attachment.
Mix on low speed for 3 to 4 minutes, then scrape the bottom and sides of the bowl with a silicone spatula.
Beat on low speed for 3 to 4 minutes longer.
Add the soft butter, about 2 tablespoons at a time, allowing each piece to completely incorporate before adding the next.
When all of the butter has been added, mix on low speed for 10 minutes.
Turn the mixer up to medium speed and mix for 15 minutes, or until the dough pulls away cleanly from the sides of the bowl and is stretchy and somewhat shiny.
Place the dough in a large bowl and press plastic wrap directly onto its surface.
Allow the dough to proof in the refrigerator overnight, or for a minimum of 6 hours.*
Divide the dough into 24 equal portions (about 59 grams each).
Lightly flour the work surface, roll each portion into a tight ball, and flatten into a patty shape.
Place on parchment-lined baking sheets, cover loosely with plastic wrap, and allow to rise in a warm place until doubled in size (about 4 to 5 hours).
Preheat the oven to 350 degrees F.
Beat the remaining egg with about 2 tablespoons of water, and gently brush the egg wash over the tops of the buns.
Bake the buns (1 baking sheet at a time) for 20 to 30 minutes, or until the tops are completely golden brown.