Warm the Greek yogurt in the microwave for 30 seconds until slightly warm to the touch, not hot. Hot yogurt can damage the wild yeast in your sourdough starter.
In the bowl of a stand mixer fitted with the dough hook, combine the sourdough starter, warmed Greek yogurt, eggs, and sugar. Mix on low to combine.
In a separate bowl, whisk together the bread flour, baking powder, and salt. Add the dry ingredients on top of the wet mixture. Cut the softened butter into marble sized pieces and add them to the bowl.
Mix on low speed until a shaggy dough forms and the flour is fully incorporated, about 2 minutes. The dough will feel sticky at this stage, which is normal for a yogurt based sourdough.
Cover and let rest 30 minutes. This hydration rest is essential. The yogurt needs time to fully integrate with the flour, and skipping this step leads to a dough that feels impossible to work with.
After the rest, mix on low for 1 minute to bring the dough back together, then increase to medium low and knead for 6 to 8 minutes until the dough is smooth, soft, and slightly tacky. The dough should pull away from the sides of the bowl. If after kneading the dough is still sticking to the sides, add flour 10 grams at a time until it reaches the right consistency.
Use a bench scraper to pull the dough cleanly off the sides of the bowl and gather it into a ball. Place the dough in a lightly oiled bowl, cover with plastic wrap or a damp towel, and let rest at room temperature for 1 hour.
After 1 hour, perform one set of stretch and folds. With lightly dampened hands, grab one side of the dough, stretch it up, and fold it over the center. Rotate the bowl and repeat three more times. Finish with one or two slap and folds to seat the seam underneath and build tension on the top of the dough ball.
Cover and continue to rise at room temperature for about 6-7 more hours, until the dough has noticeably puffed up and feels light and airy, approximately 6-8 hours total from mixing. Timing varies based on kitchen temperature. Warmer kitchens go faster, cooler kitchens take longer.
Turn the dough out onto a lightly floured surface and weigh it. Divide into 12 equal pieces of approximately 100 to 110 grams each using a bench scraper. Roll each piece into a tight, smooth ball by tucking the edges underneath and creating surface tension.
Flatten each ball firmly with your palm to about 3.5 to 4 inches wide. This step is what gives you a proper domed burger bun shape rather than a tall dinner roll. Do not skip it and do not press the buns down after the final proof.
Place the shaped buns on two parchment lined baking sheets, spaced at least 2 inches apart. Cover with a lid or large plastic bag and proof at room temperature for 1.5 to 2 hours, until noticeably puffy. If your kitchen is cool, place the rolls in the oven with the light on for 30 - 60 minutes. Preheat the oven to 375 degrees Fahrenheit during the last 20 minutes of proofing.
Alternatively, Place in 4 rows of three across on one parchemnt lined pan. The rolls will still bake well, but you pull them apart like a store boight bun. They may require a few minutes more of baking.
Whisk the egg with the water until smooth. Brush each bun gently with the egg wash. Sprinkle with sesame seeds or everything bagel seasoning if desired.
Bake for 18 to 22 minutes, rotating the pans halfway through, until deep golden brown on top. Internal temperature should read 190 to 195 degrees Fahrenheit on an instant read thermometer.
(Optional)Brush the hot buns immediately with melted butter the moment they come out of the oven. This soaks into the crust and adds one more layer of richness. Let cool on a wire rack for at least 15 minutes before slicing.