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Divide the full recipe of brioche dough in half. You’ll be working with one half the dough at a time; set the other half aside, covered, and refrigerate it.
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To make Brioche Nanterre: Divide your half-portion of dough into eight balls, about 45g each. Place four rows of two balls each into a lightly greased 8 1/2" x 4 1/2" loaf pan or 9" x 4" x 4" pain de mie pan (no lid needed).
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Cover the pan and let the loaf rise for 75 minutes, or until it’s expanded nicely. Toward the end of the rising time, preheat the oven to 375°F.
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Brush the loaf with the beaten egg and water. Bake the bread for about 28 minutes, until it’s a deep golden brown. Remove it from the oven and after a couple of minutes turn it out onto a rack to cool.
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Store any leftover bread, well wrapped, at room temperature for several days; freeze for longer storage.
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To make brioche breakfast sandwiches: Preheat the oven to 375°F. Lightly grease a baking sheet, or line it with parchment.
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Starting with one-half of the original amount of brioche dough, divide it into four pieces, about 90g each.
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Round each piece into a tight ball, and space the balls on a lightly greased or parchment-lined baking sheet. Cover them lightly with greased plastic wrap or a reusable cover, and let them rise for about 75 minutes, until they’re quite puffy.
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Working with one piece of dough at a time, gently insert the forefinger and middle finger of your hand into the center of the dough (don’t penetrate the bottom) to create an opening, pulling outward so the center is about 2 1/2” wide. There should be a thin layer of dough in the bottom, and an even rim of dough around the perimeter.
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Brush the perimeter of each round with the egg/water; you should have some left over from the Brioche Nanterre, if you've made that.
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Spread a thin layer of crème fraîche or sour cream into the bottom of each roll, then add a thin layer of the mushrooms or chopped, cooked bacon.
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Crack 1 egg into each roll, carefully centering it atop the mushrooms. Sprinkle with salt and pepper to taste, then top with grated Parmesan cheese.
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Bake the sandwiches for about 14 minutes, until the egg is barely set. Remove them from the oven and serve immediately.
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Store any leftovers, well wrapped, in the refrigerator for a day or so; freeze for longer storage.