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To make the tangzhong: In a small saucepan, combine all of the ingredients and whisk until no lumps remain.
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Place the saucepan over low heat and cook the mixture, whisking constantly, until thick and the whisk leaves lines on the bottom of the pan, about 3 to 5 minutes.
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Remove from the heat and transfer to a large mixing bowl, the bowl of a stand mixer, or the bucket of a bread machine (whatever you plan to knead the dough in).
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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. Add the cold milk, then the flour and remaining ingredients to the mixing bowl in the order listed; the heat from the tangzhong will help to warm the cold milk.
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Mix — by hand, on low speed of a stand mixer with the dough hook attachment, or in a bread machine set to the dough cycle — to bring the dough together. Next, knead the dough until it’s smooth, elastic, and tacky. This will take up to 15 minutes by hand, 10 to 12 minutes on medium-low speed of a mixer, or the length of the dough cycle in a bread machine.
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Shape the dough into a ball, and let it rest in a lightly greased bowl, covered, for 1 to 1 1/2 hours, until puffy but not necessarily doubled in bulk.
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To make the filling: In the bowl of a food processor, pulse the walnuts, flour, sugar, and salt until combined.
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Add the egg white, maple cream, and maple flavor and pulse until a smooth paste forms. Cover and set aside until you’re ready to assemble the bread.
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To assemble the maple milk bread: On a clean work surface, gently deflate the dough and roll it to a 20" x 8" rectangle, with the long side facing you.
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Dollop the filling over the surface of the dough and use a small offset spatula to spread it evenly, leaving a 1" border bare along the long edge that’s further away from you. Roll the dough toward the uncovered edge, pinching to seal the seam.
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Bring the ends of the log together to form a ring, pinching and sealing the place where they meet to enclose the filling. Flatten the ring slightly so that it forms an 8" oblong.
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Transfer the dough to a greased 9" x 4" Pullman pan. Settle the ring into the bottom of the pan with its two sides next to one another (not on top of each other), and the original long seam facing down.
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Press the ring gently with the palm of your hand to encourage the dough to reach the corners of the pan. Grease the lid, then cover and let rise until the loaf is 1" from the top of the pan, about 40 to 50 minutes. Toward the end of the rise time, preheat the oven to 350°F with a rack positioned in the center.
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Bake the bread, with the lid on, for 36 to 42 minutes. Remove the lid to check for doneness: It should be golden brown and a digital thermometer inserted into the center should read at least 190°F.
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Remove the bread from the oven and turn the oven off. Turn the bread out of the pan and place it directly on the oven rack; let it rest in the turned-off oven for an additional 5 minutes to help the bread brown on all sides.
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Remove the bread from the oven and allow it to cool completely on a rack before slicing.
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Storage information: Store any leftover Swirled Maple Milk Bread in an airtight container for up to 5 days; freeze for longer storage.