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To make the stiff levain: In the bowl of a stand mixer, use the flat beater attachment to combine the ingredients, then switch to the dough hook to mix on low speed until smooth. Cover the bowl and set it aside at room temperature (68°F to 72°F), for 4 to 6 hours, until doubled or tripled in size.
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To make the dough: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. To the bowl of the mixer with the risen levain, add the flour, honey, sugar, salt, and yeast. Mix with the dough hook on low speed for about 2 minutes; the dough will not have come together and will look very clumpy and dry. (Note: Kneading by hand is not recommended for this recipe.)
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Add the water to the bowl and continue mixing on low speed until the water is completely absorbed, 1 to 2 minutes.
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In a separate small bowl, lightly beat the egg yolks and whole egg. Add half of the beaten egg mixture to the dough, allowing the dough to absorb the eggs before adding the remaining egg mixture. The dough will look ragged and curdled at first. Keep mixing on low until all the egg is absorbed and the dough starts to homogenize, 10 to 15 minutes.
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Staying on low speed, add the oil. Allow the dough to start absorbing the oil, then scrape down the bowl and continue mixing on low for a few more seconds. Slowly increase the speed to high and keep mixing the dough until it’s smooth and elastic, clings to the hook, and cleans the side of the bowl, 5 to 8 minutes. Add the raisins and mix on low speed until just incorporated.
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Gather the dough into a rough round in the mixing bowl, then transfer it to a large, lightly greased mixing bowl or container. Cover it and let it rise at room temperature for 1 hour. Refrigerate the covered dough overnight, 8 to 16 hours.
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To shape the challah: The next day, remove the dough from the refrigerator. Working on a lightly floured work surface, divide it into 4 equal parts (about 240g each without raisins; 278g with raisins) and shape it into rounds. Allow the rounds to rest, covered with plastic or a clean towel, for 10 to 15 minutes.
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On a lightly floured work surface, start to roll out the dough to form tight logs, working with both hands from the center out until you have ropes about 18" long. (If the dough starts to shrink back, let it rest, move to the next piece, and return to that rope later. Stretching the dough progressively works best.)
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Working on a piece of parchment, braid the dough into a four-strand round (this video will show you how). Transfer the parchment to a baking sheet. Brush the loaf with a thin layer of egg wash. Cover and let rise at room temperature until doubled in size, 2 1/2 to 3 hours. Towards the end of the rise time, preheat the oven to 350°F with a rack in the center.
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When the challah has risen, give it another brush of egg wash. Bake the sourdough challah for 25 minutes; if the loaf is getting too dark, tent it loosely with aluminum foil. Bake for an additional 15 to 20 minutes, until the loaf is mahogany colored and sounds hollow when you tap on the bottom; the internal temperature should be at least 190°F.
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Let the sourdough challah cool on the baking sheet for at least 40 minutes. As it cools, make the glaze.
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To make the glaze: In a small saucepan, bring all of the ingredients to a boil, stirring lightly to dissolve the sugar. Remove from the heat and let cool. (The glaze can be prepared in advance and kept in the refrigerator for up to 1 week.)
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Just before serving, brush the glaze on the challah. Slice or tear and enjoy. Turn leftovers into French toast or bread pudding.
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Storage instructions: Store challah in a paper bag at room temperature for 3 to 4 days. Reglaze again, if desired. For longer storage, freeze the loaf well wrapped and unglazed. To reheat, loosely wrap with foil and warm up in a 350°F oven. Glaze before serving, if desired.
