Pillowy, pull-apart rolls with just enough sourdough character to be interesting without overpowering the butter you're going to put on them. An enriched dough with milk and egg, so it handles differently than a lean loaf. Give the shaped rolls a proper second rise and they'll be tall and tender.
In a large bowl, whisk together the warm milk, melted butter, egg, and sugar. Add the starter and mix to combine. Add the flour and salt and mix until a shaggy dough forms.
Turn the dough out and knead 8 to 10 minutes, either by hand or in a stand mixer with a dough hook, until it's smooth, elastic, and only slightly tacky.
Place the dough in a lightly oiled bowl, cover, and let it rise at about 78°F for 3 to 4 hours, until nearly doubled. Enriched dough rises more slowly than a lean loaf, so give it the full time, longer in a cool kitchen.
Turn the dough out and divide it into 15 equal pieces, about 55g each. Shape each into a tight ball by cupping your hand around it and rolling it against the counter. Arrange in a buttered 9x13-inch pan, evenly spaced.
Cover loosely and let the rolls rise at room temperature 1.5 to 2 hours, until puffy and touching each other. If you'd rather bake fresh in the morning, cover the shaped rolls and refrigerate overnight instead, then let them sit at room temperature about an hour before baking.
Preheat the oven to 375°F. Bake 20 to 25 minutes, until deep golden brown on top and cooked through. Brush with melted butter the moment they come out of the oven. Serve warm.
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