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To brown the butter: Have a clean, fine-mesh strainer and heatproof bowl on hand. Place the butter in a small saucepan and cook over medium heat until melted.
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Reduce the heat to medium-low and simmer until the butter separates and the solids start to smell nutty and turn brown, like the color of maple syrup (about 5 minutes).
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Strain the butter into the bowl, discarding the strained solids, and set it aside to cool to room temperature.
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To make the dough: In the bowl of a stand mixer fitted with a paddle, combine the water, milk, and egg. Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Add the flour, yeast, sugar, and salt. Mix on medium speed until a soft dough begins to form.
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Scrape the bowl, switch to the dough hook, and continue to knead for 8 to 10 minutes. The dough should be soft and stretchy.
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Turn the dough out onto a lightly floured work surface and knead briefly by hand into a smooth ball. Place in a greased bowl, cover, and let rest at room temperature for 1 1/2 hours.
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Lightly flour your work surface. Turn the dough out and press it into a rectangle. Divide it into eight portions, each weighing 3 3/4 to 4 ounces.
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Shape each piece into a ball. Brush the tops with some of the browned butter and place on a buttered piece of parchment.
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Cover and let rest at room temperature for 30 minutes.
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Brush the inside of a 9" springform pan with browned butter.
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Generously brush a second sheet of parchment with browned butter. Place a ball of dough on the buttered parchment and brush the top with more browned butter. Use your hands to flatten and carefully stretch it out to a 10" square. Don't worry about tearing the dough (it will end up on the inside).
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Brush more browned butter over the square, then fold the right side over to the center. Fold the left side over the top, making a trifold (letter fold). Brush with more butter and roll the rectangle up into a sausage shape, starting with the short side closest to you.
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Place seam side down on the first piece of parchment. Repeat with the remaining balls of dough.
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Using a sharp knife, cut each roll in half across the middle and place, cut side down, in the prepared pan. Place 10 coils around the outside of the pan, a ring of five more inside those coils, and the last one in the center. Cover the pan and let rise at room temperature for 45 minutes, until puffy-looking.
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Towards the end of the rising time, preheat the oven to 350°F.
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Place the springform pan on a rimmed baking sheet and bake the rolls for 30 minutes. Reduce the oven temperature to 325°F and continue baking for 10 or 15 minutes more. Check between the coils for doneness.
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Remove the rolls from the oven and brush them with more browned butter.
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Let the rolls cool completely before removing from the pan to serve.
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Store rolls, covered, on the counter for up to three days; freeze for longer storage.