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Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Combine flour with the rest of the dough ingredients, and mix and knead — by hand, stand mixer, or bread machine — to form a soft, smooth dough, adding more flour or water if necessary.
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Transfer the dough to a lightly greased bowl, cover, and allow it to rise until puffy though not necessarily doubled in bulk, 60 to 90 minutes, depending on the warmth of your kitchen.
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Lightly grease a 13" pain de mie pan.
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Turn the dough out onto a lightly greased work surface and roll it into a 10" x 20" rectangle.
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Spread a thin layer of the egg over the entire surface of the dough then sprinkle with an even layer of bread topping, leaving a 1/2"-wide strip uncovered along one short edge.
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Starting with the filling-coated short edge, roll the dough into a log. Place the log in the pain de mie pan, seam-side down.
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Cover the pan with a lightly greased piece of plastic (or your favorite reusable wrap) and let the bread rise for 60 to 90 minutes, or until it's puffy; it should have risen to within 1/2" of the rim of the pan.
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Preheat the oven to 350°F during the final 15 minutes of rising.
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Remove the plastic and put the cover on the pan. Bake the bread for 25 minutes. Remove the pan from the oven, carefully remove the lid, and return the bread to the oven to bake for an additional 10 to 20 minutes, or until golden brown and a digital thermometer inserted into the center registers 190°F.
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Remove the bread from the oven, allow it to cool for 10 minutes, then turn it out of the pan onto a rack to cool completely
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Store, wrapped, at room temperature for four days; freeze for up to three months.