Combine the warm milk, yeast, and 1/4 cup sugar in a large bowl. Let sit 5 to 7 minutes until foamy, then mix in the salt, softened butter, egg, and flour until a shaggy dough forms.

Knead on a floured surface for 8 to 10 minutes until smooth and slightly tacky. Place in an oiled bowl, cover, and rise for 1 hour until doubled.

Stir the 1/2 cup sugar and cinnamon together in a small bowl to make the filling.

Roll the dough into a 9x16-inch rectangle, brush with the melted butter, scatter the cinnamon sugar evenly over the surface, roll into a tight log, and place seam-side down in a greased 9x5-inch loaf pan.

Cover loosely and let rise 45 minutes until the dough crowns 1 inch above the pan. Heat oven to 350°F (175°C), brush the loaf with egg wash, and bake 35 to 40 minutes until deep golden brown and the center reads 190°F to 195°F (88°C to 90°C).

Rest in the pan for 10 minutes, turn out onto a wire rack, slice thickly with a serrated knife, and serve warm with the swirl facing up.