Prepare the levain in a pint jar or bowl. Mix flour, water, and sourdough starter, loosely cover, and let develop in a warm place for 3 hours. 

Preheat oven to 425°F. Scrub the potatoes and prick all over with a fork. Place on parchment-lined sheet pan and bake for 1 hour, until internal temperature is around 200°. Let cool, then scoop flesh from skins and rice or mash until smooth. Cool completely.

Cut butter into tablespoon-sized pats and allow to come to room temperature.

Once levain has matured, mix the dough: In an electric mixer with dough hook, combine flour, milk, eggs, sugar, salt, and mature levain. Scrape the bowl’s sides, increase speed to medium, and mix for 5 minutes; the dough will begin to cling to the hook and pull away from the bowl. Let rest 10 minutes.

Mix dough on medium for a couple more minutes. Then, with mixer running, add butter a pat at a time, waiting until each is fully incorporated before adding the next. When butter is incorporated, add half of the riced potatoes and mix until combined, then add the remaining potatoes. 

Remove dough hook and cover bowl. Allow to ferment for 3–4 hours, with 4 sets of stretch-and-folds occurring 30 minutes apart.

When fermentation begins, set a timer for 30 minutes. After the allotted time, either remove dough from bowl and work on the counter in a “slap and fold” technique, or work dough in the bowl using a “stretch and fold” technique, pulling the dough from bottom to top at the clock positions of noon, 3, 6, and 9. Repeat 3 times at 30-minute intervals, then allow the dough to rest for the remaining 1–2 hours.

Once dough is puffy and shows signs of active fermentation, transfer to a work surface and divide in half with a bench scraper. Further divide each half into 13 pieces, sprinkle a little flour over each, and form each into a taut ball by tucking dough from the top of the ball down and underneath, working your way around. 

(If making cloverleaf rolls, divide each half into 12 then divide each piece in thirds...roll those little babies!)

Arrange in 2 round (or square) cake pans or 1 large rectangular pan, ungreased. Cover and, if baking same-day, proof for 2–3 hours at room temperature until puffy. If preparing over 2 days, place in the refrigerator overnight then proof at room temperature the next day until puffy.

About 30 minutes before the end of the proof, preheat oven to 425°F. Whisk the egg and milk to form a wash and brush over the roll tops. Bake 10 minutes, lower heat to 350°, and continue baking for 15 minutes, until browned and the internal temperature reaches about 200°.

For cloverleaf rolls in muffin pan: bake about 5 minutes less.

For burger buns, bake on a parchment-lined sheet pan, and keep the temp high, 425°F, for whole time, 20–25 minutes.