Warm the Greek yogurt and milk in the microwave for about 30 seconds, just enough to take the chill off. You want it slightly warm but not hot. If the yogurt is too warm it can weaken your starter when they are mixed together.
In a large bowl whisk together the bread flour, baking powder, salt, and sugar. In a separate bowl whisk together the sourdough starter, warmed yogurt, milk, and egg until smooth. Pour the dry mixture into the wet ingredients then throw walnut size chunks of the butter on top. Mix until a cohesive dough forms. Let rest 10 minutes.
Two options: Knead the dough on a clean surface for 6 to 7 minutes, OR mix on medium speed with a stand mixer dough hook, until the dough smooths out and becomes elastic. This dough will feel sticky at first because the Greek yogurt releases moisture slowly and differently than water would. Do not add extra flour yet. Give it the full knead time. The bread flour will absorb the yogurt and the dough will tighten up considerably.
Scrape the dough into a ball after removing the dough hook.
Place the dough in a lightly oiled bowl and cover. After the first 30 to 45 minutes, perform a set of stretch and folds until the dough begins to tighten up. Perform one slap and fold to get the seam in the dough to the bottom of the bowl. Creating a smooth rounded top. Then cover and leave it undisturbed. Bulk ferment until the dough is 50 to 75 percent larger in volume, the surface looks domed and jiggly when you gently shake the bowl, and bubbles are visible around the edges. This takes 6 to 10 hours at room temperature depending on your kitchen temperature. For a longer overnight bulk ferment, a room temperature of 66 to 68 degrees Fahrenheit is ideal.
Turn the dough out onto a lightly floured surface. Roll it gently into a rectangle roughly the width of your loaf pan, about 10 inches wide and 8 inches long. Roll the dough up tightly from the short end, pulling gently and rolling (Like you roll a sleeping bag) creating surface tension as you go. I like to pull the ends in every few rolls to keep them well shaped. Pinch the seam firmly all the way across and place the shaped loaf seam side down in a parchment lined or lightly greased 10x5 loaf pan. (see the blog post for a video guide)
Cover with a plastic bag (make sure the bag is inflated so the plastic doesn't touch the dough) and let the shaped loaf proof at room temperature for 1.5 to 2.5 hours, until the dough crowns about 1/2 inch above the rim of the pan. However the poke test is the most accurate way to tell if your dough is ready to be baked. If your kitchen runs cool, you can place the pan in the oven with just the oven light on, for a short period to create a gentle proofing environment. You don't want to go above 80 degrees fahrenheit.
Remove the loaf from the oven if proofing inside and preheat to 375 degrees Fahrenheit. While the oven heats, whisk together the egg and water to make the egg wash and brush it gently over the surface of the loaf.
Bake for 38 to 42 minutes until deep golden brown. Check the internal temperature with an instant read thermometer. You are looking for 190 to 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 melted 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.