In a large bowl, combine warm milk, water, honey, and active sourdough starter until evenly mixed.
Add the whole wheat and white flours and stir with a wooden spoon until a rough, shaggy dough forms. Cover and let rest for 45–60 minutes to autolyse.
Sprinkle the salt over the dough and gently mix it in.
Add the softened, cubed butter and stir with a wooden spoon. Wet your hands and knead or use stretch-and-fold motions until the butter is fully incorporated and the dough is smooth and elastic.
Over the next hour, perform 2–3 sets of stretch-and-folds, spaced about 20 minutes apart. This develops a smooth, elastic dough without traditional kneading.
Place the dough in a lightly oiled bowl, cover with plastic wrap or a damp towel, and let rise until about 50% larger (3–6 hours).
Turn the dough out onto a lightly floured surface. Divide into 12 roughly even pieces and shape each into a tight ball by tucking edges underneath to create surface tension.
Arrange the dough balls in a greased 9x13-inch baking pan so they are just touching. Cover and let rise until puffy and airy—don’t let them fully double. Preheat oven to 180°C (350°F) when they’re nearly ready.
Whisk together 1 egg and 1 tablespoon milk. Gently brush the tops of the rolls with the egg wash.
Bake for 20–25 minutes, until golden on top. Let cool for 15–20 minutes before serving.