Warm the Greek yogurt in the microwave for 30 seconds until it is slightly warm to the touch but not hot, because hot yogurt can damage the wild yeast in your starter while cold yogurt will drag the bulk ferment out by hours.

In the bowl of a stand mixer fitted with the dough hook, combine the sourdough starter, warmed Greek yogurt, eggs, and sugar, and mix on low just to bring everything together.

In a separate bowl, whisk together the bread flour, baking powder, and salt, then 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 exactly what a yogurt based sourdough is supposed to do.

Cover and let the dough rest for 30 minutes without touching it. This hydration rest is essential, since the yogurt needs time to fully integrate with the flour, and skipping it leaves you with 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 and pulls away from the sides of the bowl. If it is still clinging to the sides after the full knead, add flour 10 grams at a time until it comes together.

Use a bench scraper to pull the dough cleanly off the sides of the bowl and gather it into a ball, then place it in a lightly oiled bowl, cover with plastic wrap or a damp towel, and let it rest at room temperature for 1 hour.

After 1 hour, perform one set of stretch and folds with lightly dampened hands, grabbing one side of the dough, stretching it up, and folding it over the center, rotating the bowl and repeating three more times. Finish with one or two slap and folds to seat the seam underneath and build tension across the top of the dough ball.

Cover and continue to rise at room temperature for another 6 to 7 hours, until the dough has puffed and domed rather than sitting flat across the top, jiggles as a single soft mass when you gently shake the bowl, and shows bubble activity along the edges where it meets the bowl. You are looking for roughly a 50 to 75 percent increase rather than a full doubling, which is about 6 to 8 hours total from mixing. Warmer kitchens move faster and cooler kitchens take longer, so watch the dough rather than the clock.

Turn the dough out onto a lightly floured surface and weigh the entire mass, then divide that number by six to get your portion weight, which will land right around 200 to 205 grams per piece. Weighing the whole dough and dividing is more accurate than aiming for a fixed number, since starter hydration, yogurt brand, and flour left behind on the counter all shift the total slightly from batch to batch.

Preshape each piece into a log about 6 inches long. Press the portion into a small rectangle, fold the top down to the center and press, fold it over on itself once more, and roll it closed. Seal the seam lightly rather than tightly, because the point of the preshape is to organize the dough and leave it something to give at final shaping. See blog post for other shaping options.

Roll the log into a rectangle roughly 7 to 7 1/2 inches wide and about 5 inches deep, with the long edge running parallel to the counter edge in front of you. Aim for about a third of an inch thick and keep the corners squared rather than rounded, since rounded corners are what give you skinny tapered ends later. Your length comes from how wide you roll this rectangle rather than from stretching the log afterward.

Starting at the far long edge, fold about an inch of dough down onto itself and press that seam firmly into the sheet below it. Keep rolling toward yourself one turn at a time, pressing each new seam down before you make the next one, and letting your fingertips pull back very slightly so the log stays taut without stretching. When you reach the near edge, pinch the final seam closed along its entire length including right at the two ends.

Give the log one or two passes under flat palms to even out any thick spots, easing the pressure off through the last inch and a half at either end so you finish with blunt hoagie ends rather than tapered points. The finished roll should measure about 7 1/2 inches. Place it seam side down on the prepared pan and move to the next one.

Space the rolls 2 1/2 to 3 inches apart, since they spread outward more than a burger bun does. Do not press them flat. Cover with a lid or a large plastic bag and proof at room temperature for 1.5 to 2 hours. They are ready when they look visibly puffy and rounded with the sides starting to bulge, and when a fingertip pressed gently into the side leaves a dent that fills back in slowly and only partway. Aim for a 50 to 75 percent increase rather than a full double, since anything you give away on the final proof comes straight out of your oven spring.

Preheat the oven to 375 degrees Fahrenheit. Pull the proofing cover off for the last 15 minutes so any condensation on the tops has a chance to dry off before the wash goes on, since a damp surface is one of the main things that holds back browning. Do not chill the rolls before baking.

Whisk the egg yolk and cream together until smooth. Apply the wash first and score second, since egg brushed into an open cut can set and glue the score shut before the roll finishes expanding. Lay the wash on gently with a folded paper towel or a soft pastry brush, using light strokes rather than pressure, and carry it down the shoulders on each side so you do not get a color line where the wash stopped. One thin even coat is all you need, since a heavy coat bakes up blotchy.

Let the wash sit for about two minutes so it tacks up, then score. Make three or four short diagonal cuts across the width of each roll, each about 2 inches long, angled roughly 45 degrees off the long axis, spaced about an inch and a half apart, and stopping about an inch in from each end. Hold the blade close to straight up and down rather than angled, go about three eighths of an inch deep, and make one confident pass per cut rather than several careful ones. Wipe the blade between rolls so egg residue does not build up on the edge. See the scoring note below for other options.

Bake uncovered for 26 to 30 minutes, rotating the pans halfway through, until deep golden brown. The internal temperature should read 190 to 195 degrees Fahrenheit on an instant read thermometer, and the thermometer is more reliable here than the clock. Tent loosely with foil if they are getting dark before the center comes up to temperature.

Cool completely on a wire rack before slicing. A yogurt crumb is still setting for a good twenty minutes out of the oven, and slicing warm will smear it.