For the dough: Warm the milk and 1/2 cup water in a medium saucepan over low heat until it is about 100 degrees F (but no more than 110 degrees F). Remove the saucepan from the heat, and sprinkle the yeast over the 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 together 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 and slightly 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 it 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 for 4 hours or overnight to let the dough proof.
For the filling: Cook the bacon in a large nonstick skillet over medium heat until browned and crisp, about 5 minutes. Transfer the bacon to a small bowl, and remove all but 1 tablespoon of the fat from the skillet. Return the skillet to medium heat. Whisk the flour into the bacon fat, and cook, whisking, until light brown, about 1 minute. Slowly whisk in the milk, and continue whisking until thickened, about 1 minute more. Transfer the mixture to a medium bowl, add the egg yolk, Cheddar, chives, cooked bacon, 1/2 teaspoon salt and a few grinds of pepper and stir to combine. Let 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 buns: Butter a 9-by-13-inch baking pan with 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 a long edge facing you. Spread the filling evenly over the dough, leaving a 1-inch border on the side opposite you. 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 the buns loosely with plastic wrap, and set aside in a warm place to rise until the buns double in size, leaving no gaps in between, 1 hour 30 minutes to 2 hours.
For the topping: Stir the butter, chives 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, and return them to the oven for 5 minutes more. Let them cool in the pan for 10 minutes before serving.
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