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Combine milk and butter in glass measuring cup. Microwave on high until butter melts and mixture is just warmed to 120°F to 130°F, 30 to 45 seconds. (If you don’t own a microwave, gently heat the butter and milk together in a small skillet.) Place 1 cup (128 g) of the flour, the sugar, the egg and the salt in the bowl of a stand mixer fitted with the dough hook and mix on low speed. Slowly pour the warm milk and butter mixture into the bowl. Beat on low speed 3 minutes, stopping occasionally to scrape down sides of bowl. Touch the mixture to make sure it is just barely warm. Sprinkle yeast over top and stir. Add remaining 2 1/2 cups (318 g) flour to the mixing bowl. Continue beating on low speed until flour is absorbed, about 8 minutes. (Note: In the past, I have added more flour, about a quarter cup, but most recently I don’t add any more flour. The dough will not gather around the hook — it will look like a sticky mess — but after the two hour rise, it is surprisingly easy to work with.)
Lightly grease a large bowl with butter or oil. Transfer dough to bowl, turning to coat. Cover bowl with plastic wrap. Let dough rise in warm draft-free area until doubled in volume, about 2 hours.
Meanwhile, mix sugar and cinnamon in medium bowl. Punch down dough. Divide the dough in half. Roll each into a large rectangle about ½-inch thick. Spread 2 (or more) tablespoons butter onto each half, sprinkle with just enough cinnamon sugar to cover — you will have extra; don’t go overboard or you will have issues with the cinnamon and sugar oozing out during the baking — and roll into a log starting with the short side. Place each log seam side down on a baking sheet lined with parchment paper. Make cuts every couple of inches straight down leaving the last layer of the log intact. Gently pull the sections outward as pictured above. Let rise for about 40 to 45 minutes or until slightly puffed. Preheat the oven to 375 degrees.
Brush the loaves with the beaten egg. Bake for 15 to 20 minutes or until golden — this may take more or less time depending on the size of your loaves and on your oven. Let cool slightly. Whisk the powdered sugar and milk together. Drizzle over top. Serve.