['2 1/4 cups/315 g all-purpose flour', '2 1/4 cups/340 g bread flour', '3 1/4 tsp active dry yeast, or 1 oz/30 g fresh cake yeast', '1/3 cup plus 1 tbsp/80 g granulated sugar', '1 tbsp kosher salt', '5 large eggs, plus 1 large egg for the egg wash if making loaves', '1 cup plus 6 tbsp/310 g unsalted butter, at room temperature, cut into 10 to 12 pieces', 'stand mixer with dough hook attachment', 'two 9-by-5-in/23-by-12-cm loaf pans if making loaves', 'parchment paper if making loaves (optional)']
1. Using the stand mixer, combine the all-purpose flour, bread flour, yeast, sugar, salt, 5 eggs, and 1⁄2 cup/120 ml water and beat on low speed for 3 to 4 minutes, or until all of the ingredients have come together. Stop the mixer as needed to scrape the sides and bottom of the bowl to make sure all of the flour has been incorporated into the wet ingredients. Once the dough has come together, beat on low speed for 3 to 4 minutes longer. The dough will be stiff and seem quite dry.
2. Still on low speed, add the butter, one piece at a time, mixing after each addition until it disappears into the dough. After all of the butter has been added, continue mixing on low speed, stopping the mixer occasionally to scrape the sides and bottom of the bowl, for about 10 minutes. It is important to mix all of the butter thoroughly into the dough. If necessary, break up the dough with your hands to help incorporate the butter.
3. Once the butter has been completely incorporated, increase the mixer speed to medium and beat for another 15 minutes, or until the dough becomes sticky, soft, and somewhat shiny. It will take some time for it to come together; it will look very shaggy and questionable at the start but in time will turn smooth and silky. When that happens, increase 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 2 to 3 tbsp 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.
4. Transfer the dough to 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, expand and develop flavor) in the refrigerator for at least 6 hours or up to overnight. (At this point the dough may be frozen in an airtight container for up to 1 week. Thaw overnight in the refrigerator before using.) If you are making a brioche treat, proceed as directed in the individual recipe.
5. Line the bottom and sides of the loaf pans with parchment paper, or butter the pans liberally.
6. Divide the dough in half, and press each half into about a 9-in/23-cm square. (The dough will feel like cold, clammy Play-Doh.) Facing the square, fold down the top one-third toward you, and then fold up the bottom one-third, as if folding a business letter. Press to join the three layers. Turn the folded dough over and place it, seam-side down, into one of the prepared pans. Repeat with the second piece of dough, placing it in the second prepared pan.
7. Cover the loaves lightly with plastic wrap and place in a warm (78° to 82°F/25° to 27°C) area to proof for 4 to 5 hours, or until the loaves have nearly doubled in size. They should have risen to the rim of the pan and have a rounded top. When you poke at the dough, it should feel soft, pillowy, and light, as if it were filled with air—because it is! The loaves have finished proofing and are filled with yeast air pockets. At this point, the texture of the loaves always reminds me a bit of how it feels to touch a water balloon.
8. Preheat the oven to 350°F/180°C, and place a rack in the center of the oven.
9. In a small bowl, whisk the remaining egg until blended. Gently brush the tops of the loaves with the beaten egg. Bake for 35 to 45 minutes, or until the tops and sides of the loaves are completely golden brown. Let cool in the pans on wire racks for 30 minutes, then turn the loaves out of the pans and serve warm or continue to cool on the racks. The bread can be stored tightly wrapped in plastic wrap at room temperature for up to 3 days (if it is older than 3 days, try toasting it) or in the freezer for up to 1 month.
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