Mix the dough: In a large bowl, mix the sourdough starter, milk, eggs, honey, bread flour, and whole wheat flour until well combined and no dry flour remains. Cover and let the dough rest for 30 minutes.
Add butter and salt: Add the softened butter and salt. Knead briefly until the butter is fully incorporated and the dough is smooth and soft. Cover and let rest for 30 minutes.
Stretch and fold: Over the next 1½–2 hours, perform 2–3 gentle stretch-and-folds, about 30 minutes apart. For each set, stretch one side of the dough upward without tearing and fold it over the center. Turn the dough and repeat on the remaining three sides. Lightly wet your hands and the counter to prevent sticking.
Bulk fermentation: Place the dough in a lightly greased bowl, cover, and let rise at warm room temperature until it has increased in volume by 60–70%. This usually takes about 3 hours in my warm kitchen, but timing will vary.
Divide and shape: Gently turn the dough onto a lightly floured counter. Divide into 8 equal pieces for large rolls or 12 for smaller rolls. Shape each piece into a smooth, tight ball by pulling the edges underneath, pinching them together, and gently rolling the ball against the counter.
Final proof: Place the rolls in a greased 9 × 13-inch baking dish, leaving a little space between them. Cover and let rise until puffy, increased in volume by about 50–60%, and just touching each other. This usually takes around 2 hours in my warm kitchen.
Bake: Preheat the oven to 180°C (350°F). Bake for 20–25 minutes, until golden brown on top and cooked through.
Brush and serve: Immediately brush the tops with melted butter. Let the rolls cool for at least 15 minutes before pulling them apart and serving.