Warm the milk and eggnog to approximately 105–110°F. The mixture should feel warm, not hot.
Sprinkle the yeast over the warm milk mixture and let sit for 5–10 minutes, or until foamy.
Whisk the egg, melted butter, and vanilla extract into the yeast mixture.
Add the flour and cup granulated sugar to the bowl of a stand mixer. Mix briefly to combine.
Switch to the dough hook. Add the wet ingredients and mix on low speed until a dough forms.
Increase to medium speed and knead for 6–8 minutes, or until the dough is smooth and elastic and pulls away from the sides of the bowl.
Shape the dough into a ball.
Lightly grease a large bowl. Place the dough in the bowl and turn to coat.
Cover and let rise in a warm place for about 1½ hours, or until doubled in size.
Turn the dough onto a lightly floured surface and roll it into approximately a 12x15-inch rectangle.
Spread the cup softened butter evenly over the dough, leaving about a ½-inch border along one long edge.
In a small bowl, combine the granulated sugar and cinnamon. Sprinkle evenly over the butter.
Starting with the long side opposite the uncovered edge, tightly roll the dough into a log.
Lightly moisten the uncovered edge with water and pinch to seal.
Cut the log into 12 equal rolls.
Line a 9x13-inch baking dish with parchment paper or lightly grease it. Arrange the rolls in the pan, leaving a little space between each.
Cover and let rise for about 45 minutes, or until puffy.
Preheat the oven to 325°F.
Bake for 20–30 minutes, or until the rolls are lightly golden and cooked through.
Let the rolls cool for about 15 minutes.
To make the glaze, whisk together the powdered sugar, eggnog, and melted butter until smooth.
Drizzle the glaze over the warm rolls and serve.
Notes
Make sure the milk and eggnog mixture is warm, approximately 105–110°F, but not hot. Too much heat can damage the yeast.If the glaze is too thick, add a small amount of eggnog. If it is too thin, add additional powdered sugar.Store leftover rolls covered in the refrigerator because the glaze contains eggnog.Reheat briefly before serving, if desired.