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. Yogurt that is too hot can damage your starter when they come into contact.
In a large bowl whisk together the white whole wheat flour, bread flour, baking powder, and salt. In a separate bowl whisk together the starter, warmed yogurt, whole milk, honey, and egg until smooth, then mix in the softened butter until fully incorporated. Pour the wet mixture into the dry ingredients and mix until a soft slightly tacky dough forms.
Cover the bowl and let the dough rest for 30 minutes. This rest period is important because the Greek yogurt hydrates the flour more slowly than water does. The dough will feel stickier at first but smooths out considerably after the rest without any added flour.
After the rest, knead the dough in a stand mixer on speed 2 for 5 to 6 minutes or by hand for 8 to 10 minutes until it smooths out and becomes elastic. You are looking for a partial windowpane where the dough stretches without immediately tearing. If the dough is still unworkably wet after the rest, add bread flour 10 grams at a time, mixing fully between each addition before deciding if more is needed.
Place the dough in a lightly oiled bowl, cover tightly, and let it bulk ferment at room temperature. Perform one set of stretch and folds at the 30 to 45 minute mark, then leave it alone. This enriched dough will often fully double during bulk fermentation. Look for a full dome, a jiggly airy texture when you gently shake the bowl, and visible bubbles on the surface. Expect 5 to 8 hours depending on your kitchen temperature. Whole wheat ferments faster than white flour so start checking early.
Turn the dough out onto a lightly floured surface. To shape using the rectangle fold method, pat the dough into a rectangle roughly 14 inches wide and 10 inches long. Fold the left side into the center, then the right side into the center so the touch. Roll the dough gently just to work out any air. Starting from the bottom roll the dough up, gently tugging and pressing lightly at each revolution to build surface tension. Pinch the side and bottom seams firmly and place seam side down in a parchment lined or lightly greased 9x5 loaf pan.
Cover the shaped loaf and let it proof at room temperature for 2 to 3 hours until the dough crowns about 1 inch above the rim of the pan. Use the poke test to confirm readiness: lightly oil your finger and press about half an inch into the dough. If it springs back slowly and leaves a slight indent it is ready. If it springs back immediately it needs more time. If the indent stays completely it is approaching over proof and should go in the oven right away.
Preheat the oven to 375 degrees F.
Whisk together the egg and splash of whole milk for the egg wash and brush the top of the loaf gently with one thin even coat. Optionally, Place a second loaf pan of the same size inverted loosely over the top of your loaf pan before placing it in the oven. This controls the oven spring during the first 10 minutes and can prevent side splitting.
Bake time is about 38-42 minutes. If using the lids, bake for 10 minutes with the inverted pan on top, then carefully remove it and continue baking for 28 to 32 more minutes until deep golden brown. Honey deepens the crust color faster than sugar so keep an eye on it and tent loosely with foil if the top is browning too fast before the interior is done. Pull the loaf when an instant read thermometer inserted in the center reads 190 to 195 degrees F.
Remove from the oven and brush immediately with melted butter. Let the loaf cool in the pan for 10 minutes, then transfer to a wire rack. Cool for at least 2 hours before slicing. Cutting too early will give you a gummy crumb regardless of how well it baked.