2 1/2 cups unbleached all-purpose flour, plus more if needed
2 1/4 cups bread flour
3 1/4 teaspoons active dry yeast
1/3 cup plus 1 tablespoon sugar
1 tablespoon kosher salt
1/2 cup cold water
2 3/4 sticks unsalted butter, at room temperature, cut into 10 to 12 pieces
Using a stand mixer fitted with the dough hook, combine the all-purpose flour, bread flour, yeast, sugar, salt, water, and 5 of the eggs. Beat on low speed for 3 to 4 minutes, or until all the ingredients are combined. Stop the mixer, as needed, to scrape the sides and bottom of the bowl to make sure all the flour is incorporated into the wet ingredients. 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.
With the mixer on low speed, add the butter, 1 piece at a time, mixing after each addition until it disappears into the dough. 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 the butter to be thoroughly mixed 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 until the dough becomes sticky, soft, and somewhat shiny, another 15 minutes. It will take some time to come together. It will look shaggy and questionable at the start and then eventually it will turn smooth and silky. Turn the 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 1 piece.
Put the dough in a large bowl or plastic container and cover it with plastic wrap, pressing the wrap directly onto the surface of the dough. Let the dough proof (that is, grow and develop flavor) 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.