For the levain: Combine the all-purpose flour, whole-wheat flour, water and starter in a medium bowl and mix to combine. Cover and let sit at room temperature until bubbly and increased in volume by about 20 percent, about 8 hours.
For the dough: Add the all-purpose flour, water and whole-wheat flour to the bowl of a stand mixer fitted with a dough hook. Weigh out 7 ounces of the levain (about 1 cup) and add to the mixer (discard any remaining levain). Mix on low until a shaggy dough forms. Cover the bowl and let sit at room temperature for 30 minutes.
Add the salt to the dough and mix on medium speed for 2 minutes; the dough should feel a bit sticky and wet but look smooth and very stretchy. Scrape the dough into a plastic or glass container about twice its size. Cover and let sit at room temperature until puffed, about 45 minutes.
Put the dough on a floured work surface, flatten it gently, then fold in thirds like a business letter. Transfer it back to the container. Cover and let sit at room temperature for 45 minutes.
Repeat the folding process one more time, transfer back to the container, cover and let sit at room temperature for another 45 minutes. After the third 45-minute rest, the dough should feel soft and pillowy; when pressed with a finger, it should leave an indentation that begins to spring back.
Gently scrape the dough onto a lightly floured work surface. Divide in half using a bench scraper. Shape each piece loosely into a round by gently tucking the edges under. Cover with a kitchen towel and let rest for 30 minutes.
Drape 2 kitchen towels into 2 medium (8-inch) bowls and dust very generously with flour. Dust the top of the dough rounds very lightly with flour and flip over using a bench scraper. Flatten one piece of dough into a circle, then fold the left and right sides of the dough over the center. Keeping tension in the dough, begin folding down from the top edge in 3 segments, sealing the bottom edge with the last fold. Loosely cup the dough and, using your hands and the tension between the board and the dough, pull the bread very slowly towards yourself while creating a taut ball. Gently flip the dough ball over and into one of the prepared bowls. Cover loosely with another kitchen towel. Repeat the process with the second piece of dough. Place both in a warm place to rise until about 1 1/2 times their size, 2 to 2 1/2 hours.
About 20 minutes before baking, position an oven rack in the lower third of the oven, put a 4-quart Dutch oven and lid on the rack and preheat to 500 degrees F. When preheated, carefully transfer the hot Dutch oven to a heatsafe surface with oven mitts (leave the lid in the oven). Flip one round of dough over into the Dutch oven. Using a sharp knife, scissors or bread lame, score the top of the bread in a cross or desired pattern. Return to the oven, immediately cover with the lid and bake for 20 minutes. Lower the temperature to 425 degrees F, uncover and bake until dark brown, about 10 minutes more. Transfer the bread to a cooling rack to cool completely. Return the empty Dutch oven to the oven, raise the temperature to 500 degrees F and let heat for 10 minutes before repeating the baking process with the second dough round.
You can buy sourdough starter from your favorite bakery, or order it online from a baking supply company. If you have the time, you can also make your own starter from scratch. (Search for How to Make Sourdough Starter on FoodNetwork.com.)
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