Whisk the warm water, warm milk, yeast, and 2 Tablespoons of sugar in the bowl of a stand mixer fitted with the dough hook, then cover loosely and let it stand for 5 to 10 minutes. The surface should look foamy and frothy, which tells you the yeast is awake and ready to work.
Add the remaining sugar, the softened butter pieces, 1 cup of flour, and the salt, then mix on low speed for 30 seconds and scrape the bowl. The mixture will look rough at first, and a few butter bits may still be visible.
Mix in the second cup of flour on medium speed, then add the remaining flour and continue mixing until the dough gathers and starts pulling away from the sides, about 2 minutes. The dough should feel soft and a little tacky, not wet and loose, and you can add more flour 1 Tablespoon at a time only if needed to make it workable.
Knead the dough for 8 to 10 minutes in the mixer or by hand on a lightly floured surface, adding flour by the teaspoon only if it becomes too sticky to handle. When ready, the dough should feel elastic, slightly tacky, and slowly spring back when poked.
Transfer the dough to a lightly greased bowl, turn it once to coat, cover, and let it rise in a warm spot for 1.5 to 2 hours. It should become puffy and roughly double in size.
Grease a 9×5-inch loaf pan and stir together the 1/4 cup sugar and cinnamon in a small bowl. Having the pan and filling ready makes shaping much smoother once the dough is rolled out.
Press down the risen dough, then roll it on a lightly floured surface into an 8×20-inch rectangle. The shape does not need to be perfectly sharp at the corners, but aim for an even thickness so the loaf bakes evenly.
Brush the dough with the beaten egg white, sprinkle the cinnamon sugar over the surface, and leave a 1-inch border uncovered around the edges. The coating should look even from end to end so every slice gets a good swirl.
Roll the dough into an 8-inch log, place it seam-side down in the prepared pan, and sprinkle any escaped filling over the top instead of tossing it. The loaf should sit snugly in the pan with the seam tucked underneath.
Cover the pan and let the shaped loaf rise for 1 hour. It should rise to about 1 inch above the rim of the pan and look noticeably lighter and puffier.
Set an oven rack in a lower position and heat the oven to 350°F (177°C). Baking the loaf lower in the oven helps the top color more gently.
Brush the top of the loaf with the melted butter and bake for 35 to 45 minutes, tenting loosely with foil halfway through if the top is browning too quickly. The bread is ready when the crust is golden brown and the loaf sounds hollow when gently tapped.
Cool the bread in the pan for a few minutes, then remove it and finish cooling on a rack for at least 10 minutes before slicing. If you wait longer before cutting, the slices stay neater and the center is less likely to feel gummy.