Scald the milk in a small saucepan. Stir in the sugar to dissolve. Let cool to lukewarm. Pour the warm water into the bowl of a stand mixer or a large bowl; sprinkle with yeast and stir until dissolved.
Add the milk mixture, sourdough starter, salt, and 3 cups of the flour. Mix with the dough hook until smooth. Cut the butter into pieces and incorporate. Add an additional 1 cup flour to make a smooth dough. (This dough will be STICKY!!)
Turn dough out onto a lightly floured surface and knead until smooth and elastic, 2-3 minutes. Form into a ball and place into a greased bowl. Cover with plastic wrap or disposable plastic shower cap and let rise in a warm place until doubled, 1 - ¼ hours.
Punch down dough and let rest for about 15 minutes. Grease two 9 x 5-inch loaf pans. Divide the dough in half and shape into loaves.
Transfer the dough to the loaf pans. Spray some plastic wrap with cooking spray, so that it doesn't stick, and loosely cover the loaves. Let rise in a warm area until doubled, about 1 - 1 ¼ hours.
Make a cut on the top, lengthwise of each loaf with a sharp serrated knife or a bread lame.
Use a pastry brush to brush the tops gently with an egg wash.
Preheat the oven to 400 degrees F. Bake for about 40 minutes, or until golden brown. Let cool for 10 minutes. Remove from the bread pans and let cool completely.