Whisk the cold milk, cold water, and sugar directly into the warm paste until smooth. This brings the temperature down. Check that it feels no warmer than lukewarm before you go on.
Whisk in the egg, then the sourdough starter, then the melted butter, until the mixture looks uniform.
Add the bread flour and salt. Mix with a dough whisk or your hands until no dry flour remains and you have a shaggy mass. It will feel sticky and loose at this stage, which is normal. Use your dough scraper to clean off the sides of the bowl and make the dough into a round in the center of the bowl.
Cover and rest 1 hour.
Perform 3 sets of stretch and folds spaced 1 hour apart, covering the bowl between each set. For each set, pull the edge of the dough up and fold it into the middle, turning the bowl as you go until you have worked all the way around about 6 times. Then put your fingers on the seam and your thumb against the side of the bowl, flip the dough over onto itself so the seam sits on the bottom, and push and tuck around the edge a few times until the ball is taut. (see the shaping video on the blog)
Let the dough bulk ferment at room temperature until it has risen by 50 to 70 percent, is domed on top, and jiggles when you move the bowl. This will take longer than a lean loaf because the sugar, fat, and dairy slow fermentation. Mine took about 7 hours at 72 degrees F.
Turn the dough out onto a lightly floured counter and pre shape into a round. Rest 15-20 minutes seam side down.
Turn the dough out onto a lightly floured counter. Slide both hands underneath the dough on opposite sides, then lift and fold those two sides up to meet in the middle, pushing and squeezing them together as you go. Pinch the seam closed along the top.
Turn the dough and repeat on the two opposite ends, folding them up and in with the same squeeze, then pinch those seams shut. You should have a cylinder the length of your pan with blunt ends rather than tapered points.
Place the dough seams side down and pull it back toward you, making the top to the dough taunt.
Place the shaped loaf seam side up into a floured proofing basket. Cover and refrigerate 8 to 14 hours.