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To make the ginger honey: Place the honey and ginger slices in a small saucepan, and bring to a low simmer for 5 minutes. Remove from the heat and cool, leaving the ginger slices in the honey. When the mixture is cool, remove the slices and mince them; they'll become part of the dough.
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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. Place 1 1/2 cups (180g) of the flour, the yeast, water, and eggs into the bowl of a stand mixer or the bucket of your bread machine. Beat at medium speed (or knead in the bread machine) until smooth. Cover the mixture and let sit for 45 minutes.
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After 45 minutes, the sponge will have developed some bubbles but not risen much; the yeast is getting a jump start.
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Add the remaining 1 1/4 cups (150g) of flour, the honey, minced ginger, and salt. Beat for 8 to 10 minutes (switch to the dough hook) or knead in the bread machine, until the dough cleans the sides of the bowl and becomes shiny and elastic.
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With the mixer or bread machine running, add the butter 2 tablespoons at a time, letting the butter disappear into the dough before adding the next chunk. Repeat until all of the butter is added.
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Cover the dough and let it rise for 1 hour. It'll be very soft at this point, and should have risen by about a third. Turn the dough out onto a lightly floured surface and gently fold it over several times (use a bench knife to scrape up any bits that stick to the table).
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Place the dough in a greased bowl, cover, and refrigerate for at least 4 hours and as long as 16 hours. The dough will firm up considerably.
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To shape the honey brioche: Divide the dough into six pieces. If making one large loaf, arrange them in a circle in a greased brioche mold, with the sixth one in the center. For individual breads, work with one piece at a time (keep the rest covered). Divide each into three strands, and braid them. Form the braid into a circle, then place into a greased 6" cake pans, or onto a parchment-lined baking sheet. Repeat with the remaining pieces of dough, and cover them all with greased plastic. Let the shaped bread(s) rise for 1 to 1 1/2 hours, until puffy-looking. After 45 minutes of rise time, preheat the oven to 375°F.
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To bake the honey brioche: Remove the plastic and bake the individual brioches for 25 minutes, or the large bread for 50 to 60 minutes, tenting with foil after the first 35. Bake until deep golden brown on the top and bottom. Remove from the oven and brush while still warm with the remaining ginger honey.
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Storage information: Store the honey brioche at room temperature for two to three days, or freeze for up to three months.