Make the Brioche:

• Pour the milk, yeast, 1 egg and 1 cup flour into the bowl of an electric mixer. Mix to combine, then sprinkle over another 1 cup flour. Let rise for 40 minutes.
• Lightly beat the 4 remaining eggs, then add these to the dough along with the sugar, salt and 1 cup flour. Place these into a mixer fitted with a dough hook and mix on low speed for 2 minutes. Add ½ cup flour and mix on medium speed for 15 minutes.
• Reduce the speed to medium-low and gradually add 1 ½ sticks of butter. Increase the speed to medium-high and beat for 1 minute, then reduce the speed to medium and beat for 5 minutes.
• Place the dough in a large, buttered bowl and cover with plastic wrap. Let rise for 2 ½ hours.
• Deflate the dough, then replace with plastic wrap and place the bowl in the fridge. Let sit for 4-6 hours, or up to overnight.
• Divide the dough in half. Place one ball of dough on a floured surface and roll the dough into an 11x13-inch rectangle. Evenly disperse half of the remaining butter onto the surface of the dough, then fold the dough into thirds, like a letter.
• Roll the dough out into an 11x13-inch rectangle, then fold into thirds again. Wrap tightly in plastic wrap and place in the fridge for 30 minutes. Repeat with the remaining dough.
• Combine the sugar and cinnamon in a bowl and set aside.
• Place one piece of dough on a floured surface and roll into an 11x13-inch rectangle. Brush the surface with the beaten egg. Sprinkle half of the cinnamon sugar onto the dough, leaving the top quarter of the dough bare. Roll the dough into a log, starting with the cinnamon sugar end and ending with the bare end. Wrap in plastic wrap and place in the freezer for 45 minutes. Repeat with the remaining dough. (Note: if you are making this for a party or a brunch, you can complete the previous steps a day in advance and start from Step 10 on the day you are serving the cake.
• Divide the 1 ½ sticks butter between a 9-inch, 7-inch and 3 ½-inch round baking tin. (I used casserole dishes and they worked great, so no need to stress if you don’t have baking tins in these sizes.)
• Unwrap the logs and slice them into 1 ½-inch thick buns, making 14 buns. Place 7 buns in the 9-inch tin, 6 buns in the 7-inch tin and 1 bun in the 3 ½-inch tin. Let the buns rise at room temperature for 1 ½ hours.
• Bake the buns at 350F for 35-40 minutes, until golden brown, removing the bun in the smallest tin after 20 minutes. Place a baking sheet lined with parchment paper on the rack under the cinnamon buns to catch any drips.
• As soon as the cinnamon buns are finished baking, flip them out onto a wire rack. Excess butter may drip out, so make sure to place some paper towel under the rack. Turn the buns right side up, and set slightly cool.
• To make the icing, place the cream cheese in a bowl and beat with an electric mixer for 2 minutes, until the cream cheese is fluffy. Add the confectioner’s sugar and milk and beat until combined.
• Drizzle the icing on top of each layer, then stack them to build a cake.