Mix the starter, warm milk, sugar, and bread flour. Autolyse for 30 minutes. Add salt and knead for 5-7 minutes.
Perform bulk fermentation at room temperature for 4 hours, doing a stretch and fold every hour.
Cover and cold retard in the refrigerator overnight (12-14 hours).
Pound the cold butter into a 6x6 inch square between parchment paper. Chill for 15 minutes.
Roll the dough into a 12x6 inch rectangle, encase the butter block, and pinch to seal.
Roll to 18 inches and perform 3 letter folds (turns), chilling for 45 minutes between each turn.
Roll into a 10x15 rectangle, cut into 3 strips, and braid loosely.
Place in a lined 9x5 loaf pan and proof at 75°F for 4-6 hours until puffy.
Apply egg wash. Bake at 400°F for 20 minutes, then at 375°F for 25-30 minutes.
Cool completely on a wire rack before slicing.