Remove brioche dough from freezer and let it come to room temperature. Line work surface with a large sheet of parchment paper. Lightly dust it with flour and transfer dough onto the parchment. Lightly dust dough with flour and roll into a 12× 18 rectangle that is ¼ inch thick. Arrange the rectangle so the 18-inch side is in front of you. With a small offset spatula, spread an even layer of softened butter on the dough. In a small bowl, combine sugar and cinnamon. Spread sugar mixture evenly over buttered dough. Starting from one corner, begin rolling up the long side of the dough to make a log. Tuck and tightly roll the dough over the cinnamon sugar, like rolling up a jelly roll. Work your way across to the other corner, keeping it snug as you roll, and then travel back to the other side, where you originally started. It’s important to roll the dough tightly, so the foundation is strong and the dough won’t unravel later when it is portioned and cut. Lift the parchment paper to help roll the dough up. Press on the seam to make sure it sticks. Position the brioche log so it is sitting on its seam. Join the two ends of the log together to form a ring. Pinch the seams together. Transfer the ring to a parchment lined sheet pan and place in the freezer to firm up a bit, about 10 minutes. Preheat oven to 375°. Remove chilled dough from freezer and, starting ½ inch from where the ends of the log meet, use a chef’s knife to make a cut ¼ inch from the center so that the center of the ring is still intact. Make seven more evenly spaced cuts in the same manner to yield eight cuts in total. Twist each cut dough portion clockwise so the cinnamon sugar swirl is showing. Set aside to proof in a warm place until doubled in size, about 30 minutes. Whisk egg in a small bowl to make an egg wash. Use a pastry brush to apply egg wash to entire brioche ring. Bake until dough has risen, about 12 minutes. Rotate pan and continue baking until golden brown and cinnamon sugar bubbles at the edges, about 12 more minutes. While brioche bakes, make the glaze. Allow brioche ring to cool 10 minutes. Spread glaze on top and serve.