Line a baking sheet with parchment or silicone mat.
In a small bowl, combine the melted butter, sugar, and cinnamon.
Dust the surface of the refrigerated dough with flour and cut off a 1 1/2 pound (680g, about 1/3 of above dough recipe) piece. Dust the dough piece with more flour and quickly shape it into a ball by stretching the surface of the dough around to the bottom, rotating the ball a quarter-turn as you go.
Using a rolling pin, roll the dough out to a 1/8-inch-thick rectangle, about 14x18 inches. As you roll out the dough, add flour as needed to prevent sticking.
Spread the butter mixture evenly over the dough.
Starting with the long side of the dough, roll it up into a log. Pinch the seam closed. Stretch the log until it is about 1 1/2 inches thick. Shape log into a circle, pinch the 2 ends together. Place on prepared baking sheet. Stretch the dough to make sure you have a nice, wide opening in the middle of your wreath, but leave plenty of room around the edge.
Cover loosely with plastic wrap and allow to rest at room temperature for 40 minutes.
Preheat the oven to 350°F, with a rack placed in the center of the oven.
Brush lightly with the egg wash. Make evenly spaced cuts all the way around the wreath about 1 inch apart. The cuts should go just about to the bottom of the ring, but not quite to the bottom.
Gently pull every other piece to the outside of the ring and then twist that piece to face up. Do the same with the remaining pieces, but have them face up on the inside of the ring. The ones on the inside of the ring may not lay flat on the baking sheet, which is fine.
Bake for 25 to 32 minutes, until golden brown and well set. Cool on wire rack.
Make the glaze: In a small bowl, mix together the melted butter, 2 tablespoons eggnog, and vanilla until smooth. Add the confectioners' sugar and mix until the mixture is smooth. Add more eggnog (or bourbon!), 1 tablespoon at a time, until the desired consistency is reached. Pour the glaze over the warm braid, then serve.