Combine the warm water and warm milk in a small bowl and check the temperature. It should register between 110 and 115 degrees Fahrenheit on an instant read thermometer. Sprinkle the yeast and all of the sugar over the liquid and stir gently. Let sit for 10 minutes until the mixture is foamy and fragrant. If nothing happens after 10 minutes, your yeast is dead. Discard and start over with fresh yeast.
While the yeast blooms, warm the Greek yogurt in the microwave for about 30 seconds, just enough to take the chill off. You want it slightly warm but not hot.
In a large bowl whisk together the bread flour and salt. In a separate bowl whisk together the warmed yogurt and egg until smooth. Pour the foamy yeast mixture into the yogurt mixture and stir to combine.
Pour the dry ingredients onto the wet mixture. Do not start mixing. Take teaspoons of the softned butter and place onto of the flour. Then begin to mix at speed 2 until a cohesive shaggy dough forms. Cover and let rest about 15 minutes.
Knead the dough for 5 to 6 minutes on a countertop, or mix on medium speed with a stand mixer dough hook, until the dough is smooth, elastic, and slightly tacky. This dough will feel sticky at first because the Greek yogurt releases moisture slowly and differently than water would.
Place the dough in a lightly oiled bowl, cover, and set in a warm spot. Let rise until the dough has fully doubled in size, about 1 to 2 hours depending on your kitchen temperature. Do not rely on size alone. Once doubled, perform the poke test by pressing a floured finger about half an inch into the dough. If it springs back immediately and completely, the gluten is still tight and the dough needs more time regardless of the size. You are looking for a slow partial spring back that leaves a slight indent. If it springs back slowly and only partially fills in, the dough is ready to shape.
Turn the dough out onto a lightly floured surface. Using a rolling pin, roll the dough out into a rectangle approximately 14 inches wide by 16 inches long. Position the dough so the 14 inch side is parallel to you. Fold each short side in toward the center, overlapping slightly in the middle. Starting from the edge closest to you, roll the dough up away from you, tugging gently inward and rolling as you go to build surface tension. Once rolled, pinch the seam firmly all the way across and place the shaped loaf seam side down in a parchment lined 9x5 loaf pan.
Cover the pan and let proof in a warm spot for 45 minutes to 1 hour until the dough crowns one inch above the rim of the pan.
Preheat the oven to 350 degrees Fahrenheit. While the oven heats, whisk together the egg and water or milk to make the egg wash and brush it gently over the surface of the loaf. I find using a torn piece of paper towel works better than a pastry brush.
Bake for 42 to 48 minutes until deep golden brown. Check the internal temperature with an instant read thermometer. You are looking for about 195 degrees Fahrenheit in the center of the loaf. If the top is browning faster than you would like, tent the loaf loosely with foil after the first 25 minutes and continue baking.
Remove from the oven and immediately brush the top generously with butter. Let the loaf cool in the pan for 10 minutes, then lift it out using the parchment and transfer to a wire rack.
Cool for at least 45 minutes to one hour before slicing. Cutting into an enriched loaf too soon results in a gummy crumb even when the loaf is fully baked. The interior is still setting up as it cools and the wait is worth it.