Warm the milk and butter in a small saucepan until the butter melts and the milk is lukewarm, 90 to 100°F. Do not overheat it.
Pour into a large bowl and whisk in the active sourdough starter, sugar and salt until no lumps of starter remain.
Add the flour and mix until no dry flour is left. Knead gently in the bowl for 4 to 5 minutes, adding 1 to 2 tablespoons more flour only if needed.
Cover and rest at room temperature for 30 minutes.
Bulk fermentation
With a wet hand, stretch a section of dough up and fold it over the center. Turn the bowl and repeat all the way around, then flip the dough and round it up. That is one set.
Repeat for 3 to 4 sets total, 30 minutes apart. The dough should feel noticeably stronger by the final set.
Cover and bulk rise 3 to 4 hours, until almost doubled with a bubbled surface. Time depends on kitchen temperature.
Shape and cold proof
Line two 9-inch loaf pans with parchment. Turn the dough onto a lightly floured surface and divide in two.
Pat each half into a 9-inch rectangle without pressing out the gas, roll into a tight log, and pinch the seam.
Place seam side down in the pans, cover, and refrigerate overnight for cold fermentation.
Bake
Remove from the fridge and let sit in a warm spot for 1 to 2 hours, until at room temperature and domed above the pan rim.
Preheat the oven to 360°F for 15 minutes.
Score the dough vertically. Bake for 25 to 30 minutes, until deep golden and 190 to 200°F in the center.
Cool in the pans for 10 minutes, turn out onto a wire rack, and cool completely before slicing.
Notes
Lukewarm milk only. Hot milk will damage the starter. It should feel neutral against your wrist.
Active starter is non-negotiable. It should be domed and bubbly, roughly doubled since its last feed. Discard will not work here.
Do not skip the second proof, even if you are baking the same day. Shaped loaves need 1 to 2 hours in a warm spot before they go in the oven.
Roll the log tight. Surface tension is what makes the loaf rise up rather than spread sideways.
Cool completely before slicing. Warm sourdough compresses under the knife and reads as gummy.
Freezing: slice before freezing so you can toast straight from frozen. Keeps 3 months.