1/2 cup whole milk
2 1/4 teaspoons active dry yeast (one 1/4-ounce package)
2 tablespoons sugar
4 tablespoons unsalted butter, melted, plus more for the bowl
1 large egg yolk
2 3/4 cups all-purpose flour, plus more for the work surface
1 teaspoon kosher salt
1 small clove garlic, chopped
1 small shallot, chopped
1 tablespoon fresh thyme leaves
1/2 pound cremini or white button mushrooms, ends trimmed
1/2 pound shiitake mushrooms, destemmed
4 tablespoons unsalted butter; plus 2 tablespoons, at room temperature, for the pan
1/4 cup dry white wine
2 tablespoons heavy cream
Kosher salt and freshly ground black pepper
1/4 cup grated Parmesan
4 tablespoons unsalted butter, at room temperature
1/4 cup chopped fresh flat-leaf parsley
2 tablespoons grated Parmesan
For the dough: Warm the milk and 1/2 cup water in a medium saucepan over low heat to about 100 degrees F (but no more than 110 degrees F). Remove the saucepan from the heat, and sprinkle the yeast over the liquid. Sprinkle a pinch of the sugar over the top, and set aside without stirring until foamy, about 5 minutes. Then whisk in the butter and egg yolk.
Whisk the flour, remaining sugar and salt in a large bowl. Make a well in the center, and stir in the yeast mixture with a wooden spoon until you have a thick, sticky dough. Turn the dough onto a floured work surface, and knead until soft and elastic, about 6 minutes. Shape the dough into a ball.
Brush the inside of a large bowl with butter. Add the dough ball, turning to coat lightly. Cover the bowl with plastic wrap, trace a circle the size of the dough ball on the plastic and note the time. Let the dough rise in a warm place until doubled in size, about 1 hour 30 minutes.
Turn the dough out of the bowl, and knead briefly to release excess air. Re-form the dough into a ball, and return it to the bowl. Lightly butter a large piece of plastic wrap, and lay it on the dough. Cover the entire bowl tightly with the rest of the plastic, and refrigerate it for 4 hours or overnight to let the dough proof.
For the filling: Process the garlic, shallots and thyme in a food processor until combined. Add the cremini and shiitake mushrooms, and pulse until nearly pureed (a few small chunks here and there are fine), stopping to scrape down the side of the bowl as necessary. Heat a medium nonstick skillet over medium heat, and melt 4 tablespoons of the butter. Add the mushroom mixture, and cook, stirring frequently, until it releases liquid and begins to bubble and brown, 5 to 7 minutes. Reduce the heat to low, stir in the wine and cream and bring the mushroom-cream mixture to a low boil. After 1 minute, remove the skillet from the heat, and stir in 1 teaspoon salt and a few grinds of pepper. Transfer the filling to a medium bowl, and let it cool completely. (The filling will firm up a bit, but if it's too warm, it will be difficult to form the buns.)
To fill and form the rolls: Butter a 9-by-13-inch baking pan with the 2 tablespoons softened butter. Turn the dough onto a floured work surface, and press it flat. Roll it into a 10-by-18-inch rectangle, with 1 long side facing you. Spread the filling evenly over the dough, leaving a 1-inch border on the long side opposite you. Sprinkle the Parmesan evenly over the filling. Starting from the side facing you, roll the dough up into a tight roll. Lightly brush the 1-inch border with water, and continue rolling to seal the roll. Fold the ends in just a bit, and if any spots look thicker than others, even out the roll with your hands.
Cut the roll every 1 1/2 inches with a serrated knife, to make 12 buns. Place the buns cut-side down (except for the end pieces) in the prepared pan, spacing them 1 inch apart. Cover them loosely with plastic wrap, and set in a warm place to rise until they double in size with no gaps in between, 1 hour 30 minutes to 2 hours.
For the topping: Stir together the butter, parsley and Parmesan in a small bowl; set aside.
Position an oven rack in the center of the oven, and preheat to 350 degrees F. Bake the buns until they're golden brown and the tops spring back when pressed lightly, 25 to 30 minutes. Spread the topping evenly over the buns, sprinkle with 1 teaspoon salt and return them to the oven until the butter is melted and the topping is fragrant, 5 minutes more. Let them cool in the pan for 10 minutes before serving.
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