For the dough: Add the milk and 1/2 cup water to a medium saucepan and warm over low heat until it is about 100 degrees F (but no more than 110 degrees F). Remove from the heat and sprinkle the yeast over the surface over the liquid. Sprinkle a pinch of the granulated sugar over the top and set aside without stirring, until foamy, about 5 minutes.
Whisk in 4 tablespoons of the butter, the vanilla and egg yolk until well combined.
Whisk the flour, remaining granulated sugar and 3/4 teaspoon of salt together in a large bowl. Make a well in the center of the flour and use a wooden spoon to stir in the yeast mixture until the dough becomes thick and slightly sticky. Turn the dough onto a floured work surface and knead until it is soft and elastic, about 6 minutes. Shape the dough into a ball.
Brush the inside of a large bowl with the remaining melted butter. Place the dough in the buttered bowl, turning to coat lightly. Cover the bowl with plastic wrap and use a permanent marker to trace a circle the size of the dough onto the plastic, noting the time.
Let rise at room temperature until doubled in size, about 1 hour 30 minutes.
Turn the dough out of the bowl and knead briefly on a floured surface to release excess air; reshape the dough into a ball and return it to the bowl. Lightly butter a large piece of plastic wrap and tightly cover the entire bowl. Let the dough proof in the refrigerator for 4 hours and up to overnight.
For the filling: Grease a 9- by 13-inch baking dish with some of the butter. Whisk the granulated sugar and pumpkin pie spice together in a small bowl. Turn the prepared dough onto a floured work surface and press flat. Then roll into a 10- by 18-inch rectangle, with a long edge facing you. Spread the remaining 1 1/2 sticks of butter evenly over the surface of the dough, leaving about an inch border on the side opposite you.
Dollop the pumpkin puree over the butter and spread over to cover the same area as the butter. Evenly scatter the pumpkin spice-sugar mixture over the pumpkin puree. Starting from the long side facing you, roll the dough up into a tight 18-inch-long cylinder.
Lightly brush the clean edge of the dough with water, then press it to the cylinder to seal.
Slip a long taut piece of string or unflavored dental floss under the roll, about 1 1/2 inches from the end. Lift and cross the string ends over the roll, and then pull the ends tightly in opposite directions to cut a single roll. Repeat, cutting every 1 1/2 inches, to make 12 buns. Place the buns cut-sides-down in the prepared pan, leaving 1 inch of space between them. Cover the baking dish loosely with plastic wrap and set aside in a warm place to rise until the buns double in size, about 1 hour 30 minutes.
Position the rack in the center of the oven and preheat to 350 degrees F. Bake the buns until golden brown and the tops spring back when pressed lightly, about 30 minutes. Cool in the pan for 10 minutes.
For the glaze: Whisk together the confectioners' sugar, butter, milk, vanilla and a pinch of salt in a medium bowl until smooth. Drizzle over the warm buns and serve.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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