For the dough: Combine the milk and 1/2 cup water in a medium saucepan and warm over low heat until it is about 100 degrees F (but no more than 110 degrees). 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 the butter, vanilla and egg yolk into the yeast mixture.

Whisk the flour, remaining granulated 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 the 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 the dough in the buttered bowl, turning to coat lightly. Cover the 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 30 minutes.

Turn the 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 the 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 granulated sugar and cinnamon 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 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 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 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 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 of the buns spring back when pressed lightly, about 30 minutes. Cool in the pan for 10 minutes.

For 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 the warm buns. Serve.

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