Add the milk and sourdough starter to a large bowl and whisk to combine. Whisk in the eggs, then stir in the butter. It’s okay if the butter remains in small pieces.
Add the bread flour, all-purpose flour, sugar, and salt. Mix with a wooden spoon or spatula until a shaggy dough forms and no dry flour remains.
Cover with a tea towel. Over the next hour, complete 3 sets of stretches and folds, about 20 minutes apart.