For the brioche dough: In a stand mixer fitted with the dough hook, combine the all-purpose flour, bread flour, yeast, sugar, salt, water, and eggs.Beat on low speed for 3 to 4 minutes, or until all of the ingredients have come together.Once the dough has come together, beat on low speed for another 3 to 4 minutes. The dough will be very stiff and seem quite dry.On low speed, add the butter one piece at a time, mixing after each addition until it disappears into the dough. It's important to wait until each piece is fully mixed in before adding other, or you'll have a greasy mess.Continue mixing on low speed for about 10 minutes, stopping the mixer occasionally to scrape the sides and bottom of the bowl. It is important for all of the butter to be mixed thoroughly into the dough. If necessary, stop the mixer occasionally and break up the dough with your hands to help mix in the butter.Once the butter is completely incorporated, turn up the speed to medium and beat for another 15 minutes, or until the dough becomes sticky, soft, and somewhat shiny. It will take some time to come together. It will look shaggy and questionable at the start and then eventually will turn smooth and silky.Increase speed to medium-high and beat for about 1 minute. You should hear the dough make a slap-slap-slap sound as it hits the sides of the bowl. Test the dough by pulling at it: it should stretch a bit and have a little give. If it seems wet and loose and more like a batter than a dough, add a few tablespoons of flour and mix until it comes together. If it breaks off into pieces when you pull at it, continue to mix on medium speed for another 2 to 3 minutes, or until it develops more strength and stretches when you grab it. It is ready when you can gather it all together and pick it up in one piece.
Place the dough in a large oiled bowl or plastic container and cover it with plastic wrap, pressing the wrap directly onto the surface of the dough.
Let the dough proof in the refrigerator for at least 6 hours or up to overnight. At this point, you can freeze the dough in an airtight container for up to 1 week.When you're ready to make the brioche buns: take out half of the dough. Line 10 cups of a 12 cup standard muffin tin with paper liners or generously butter and flour them.On a floured work surface, press dough into a 10 inch x 5 inch rectangle.Using a bench scraper or knife, cut dough into 10 equal 1 inch x 5 inch strips.
Cut each strip into 5 pieces. You should now have 50 squares of dough.
Place 5 squares in each prepared muffin cup. Cover with plastic wrap and place in a warm spot for about 1 1/2 hours, until the dough has risen and feels puffy and soft.
Heat oven to 350 degrees F and place rack in center of oven.
Bake for 25 to 35 minutes, until golden brown.
Let buns cool for 5 to 10 minutes on a wire rack until cool enough to handle. Meanwhile, combine sugar, spices, and salt in a small bowl.
Brush tops of buns with the melted butter.
Roll the buns in the sugar mixture to coat evenly.Buns are best served within 4 hours of baking. They can be stored in an airtight container for up to 1 day, then rewarmed in a 300 degree oven for 5 minutes.
