1. Make the dough: Lightly butter a medium bowl; set aside. In a small saucepan set over medium-low heat, heat 1/3 cup milk and 2 tablespoons butter until bubbles form along the sides of the saucepan. Remove from heat; set aside. In a 1-cup glass measuring cup, combine 1/4 cup warm water, the yeast, and a pinch sugar. Stir to combine. The mixture should begin to foam in about 5 minutes.

2. In a small bowl, lightly beat 1 egg. Slowly add milk mixture, stirring to combine. In the bowl of an electric mixer fitted with the dough hook, combine flour, a heaping tablespoon sugar, and salt. Mix on low to combine. Gradually add the milk mixture, beating on low until well combined. Add the yeast mixture, beating to combine. Stop mixer and scrape down sides of bowl as necessary.

3. Remove dough to work surface, if the dough seems sticky, check the bottom of the mixer bowl, there should be enough flour to lightly flour the board. Knead dough until it is silky and smooth and bounces back when pressed with your thumb. Transfer to prepared bowl, turning to coat in butter. Cover with a clean kitchen towel, and place in a warm spot until doubled in size, about 1 hour.

4. Preheat oven to 350 degrees.with rack in the center. Make the filling: Lightly butter an 8 1/2-by-4-by-2-inch loaf pan; set aside. In a small bowl, combine cinnamon and the remaining 2 tablespoons sugar; set aside.

5. Remove dough from bowl, and punch it down to remove air bubbles. If the dough seems sticky, lightly flour the board. If it seems dry, spritz it with a little water. Knead until silky and smooth.

6. Roll the dough into an 8 1/2-by-12-inch rectangle. Work the corners by pushing with your fingers; be careful not to tear the dough. Brush the surface of the dough with 3 tablespoons melted butter, and sprinkle evenly with the cinnamon sugar.

7. Beginning with a long side, roll the dough up into a log. Using your fingertips, be sure to push in at the seam where the dough meets the cinnamon sugar. Carefully pull the dough back to its original 8 1/2-inch width as it shrinks. When you reach the end, pinch the two layers of dough together to seal. Place the roll, seam-side down, in prepared pan. Cover, and set aside in a warm place until doubled in size, about 30 minutes.

8. Make the glaze: Whisk together remaining egg and 1 tablespoon milk. Brush over the risen dough. Bake until the crust is golden brown and the bottom sounds hollow when thumped with a finger, about 40 minutes.

9. Remove from oven, and let cool in pan for 10 minutes. Remove from pan, and cool on a wire rack for at least 10 minutes before slicing.