Warm the milk, water, and butter. Combine the water, milk, and butter in a microwave-safe container (such as a Pyrex measuring cup). Microwave on low for about 1 minute or until the mixture is warm to the touch (about 110°F). Be careful not to overheat it; the temperature should not exceed 120°F.
Combine and knead the dough. Add the warm milk mixture, eggs, sugar, flour, salt, and yeast to the bowl of a stand mixer or bread machine. Mix on low speed using the dough hook attachment until the ingredients form a dough. Increase the speed to medium and knead for 5-7 minutes until the dough is smooth and slightly sticky. Start with 3¼ cups of flour and gradually add more, if needed, until the dough pulls away from the sides of the bowl but still feels slightly tacky to the touch.
First rise. Lightly grease a clean bowl and place the dough inside. Then, cover it with greased plastic wrap and allow it to rise in a warm, draft-free spot (around 70-80°F) for 1-2 hours or until the dough has doubled in size.
Shape the dough. Once risen, turn the dough out onto a clean, lightly floured surface. Gently stretch it into a 9x24-inch rectangle, making sure an even thickness. Fold down the top corners of the dough and then roll it tightly into a log. Pinch the seams closed to seal. Place the dough seam-side down into a greased loaf pan.
Second rise. Cover the loaf with greased plastic wrap and place in a warm place to rise again for an hour until the dough has risen 1-2 inches above the edge of the pan. Brush with a beaten egg for extra shine. Towards the end of the rising time, preheat the oven to 375 degrees with a rack in the center.
Egg wash and bake. For a shiny crust, gently brush the top of the dough with a beaten egg. Bake in the oven for 30-35 minutes or until the loaf is deep golden brown and sounds hollow when tapped on the bottom. If the bread is browning too quickly, loosely cover the top with aluminum foil during the last 10 minutes of baking.
Cool and serve. Carefully remove the bread from the pan and place it on a wire cooling rack. Allow it to cool for at least 1 hour before slicing. Once cooled, the bread can be stored at room temperature for up to 3 days or frozen for up to 3 months.