1/2 cup water
1/2 cup whole milk
2 1/4 teaspoons active dry yeast (1/4 ounce package)
1/4 cup unsalted butter, melted, plus more for the bowl
1 large egg yolk
1 1/2 teaspoon pure vanilla extract
2 3/4 cup all-purpose flour (13 ounces)
1/4 cup sugar (1 3/4 ounces)
3/4 teaspoon fine salt (1/8 ounce)
1/2 teaspoon grated nutmeg
1/3 cup granulated sugar
2 tablespoons ground cinnamon
3/4 cups (1 1/2 sticks) unsalted butter, very soft, plus more for coating the pan
2/3 cups confectioners' sugar
1/3 cup sweetened condensed milk
1/4 cup melted unsalted butter
3 teaspoons freshly squeezed lemon juice
1 teaspoon pure vanilla extract
1/8 teaspoon ground cinnamon
Combine the water and milk in a medium saucepan and warm over low heat until it is about 100 degrees F (but no more than 110 degrees). Remove from heat and sprinkle the yeast over the surface over the liquid. Sprinkle a pinch of the sugar over the top and set aside without stirring, until foamy, about 5 minutes.
Whisk the butter, egg yolk and vanilla into the yeast mixture.
Whisk the flour, the sugar, salt and nutmeg in a large bowl. Make a well in the center of the flour and stir in the yeast mixture with a wooden spoon to make a thick and slightly sticky dough. Turn dough onto a floured work surface and knead until soft and elastic, about 6 minutes. Shape into a ball.
Brush the inside of a large bowl with butter. Put dough in the buttered bowl, turning to coat lightly with butter. Cover bowl with plastic wrap, trace a circle the size of the dough on the plastic, and note the time. Let rise at room temperature until doubled in size, about 1 hour 15 minutes.
Turn dough out of the bowl and knead briefly to release excess air; reform into a ball and return to the bowl. Lightly butter a large piece of plastic wrap and lay it on the dough. Cover the entire bowl tightly with plastic and proof in the refrigerator for 4 hours or overnight.
To fill and form the rolls: Butter a 9-by-13-inch baking pan. Whisk the sugar and cinnamon together in a small bowl. Turn prepared dough onto a floured work surface and press, then roll into 10-by-18-inch rectangle, with a long edge facing you. Spread the softened butter evenly over the surface of the dough, leaving about an inch border on the side opposite you. Evenly scatter the cinnamon-sugar over the butter. Starting from the long side facing you, roll the dough up into a tight cylinder. Lightly brush the clean edge of the dough with water. Press the open long edge to the dough to seal the cylinder.
Slip a long taut piece of string or dental floss under the roll, about 1 1/2 inches from the end. Lift and cross the string ends over the roll, then pull the ends tightly in opposite directions to cut a single roll. Repeat, cutting every 1 1/2 inches, to make 12 rolls. Place the rolls cut-side-down in the prepared pan, leaving 1 inch of space between them. Cover the rolls loosely with plastic wrap. Set aside in a warm place to rise until rolls double in size, about 1 hour 10 minutes.
Position the rack in the center of the oven and preheat to 350 degrees F.
Bake buns until golden brown and the tops of the buns spring back when pressed lightly, about 30 minutes. Cool in the pan for 10 minutes.
To make the glaze: Sift the confectioners' sugar into a medium bowl. Whisk in the condensed milk, butter, and lemon juice to make a smooth, slightly loose icing. Add the vanilla and cinnamon. Drizzle the icing over warm buns. Serve.
*Cook's Note: These may be refrigerated or frozen after forming. If refrigerated overnight, allow buns to come to room temperature for about 1/2 hour, then proof fully (until d