Adapted from Anne Quatrano's Summerland: Recipes for Celebrating With Southern Hospitality (Rizzoli).
Heat milk in a saucepan over low heat to 110 degrees (warm to the touch). Transfer to the bowl of a stand mixer fitted with the dough hook. Immediately sprinkle with yeast, then sprinkle 1 teaspoon sugar over yeast. Let stand until foamy, about 5 minutes. Add eggs and remaining 1/4 cup sugar. Mix on low speed until combined, about 1 minute. Add flour and salt; mix until a soft, slightly tacky dough forms, about 3 minutes. Add room-temperature butter and mix until dough is soft and slightly tacky again, about 3 minutes.
Brush the inside of a large bowl with oil. Transfer dough to bowl, cover with a clean kitchen towel, and let stand in a warm, draft-free place until doubled in size, about 1 hour.
Brush bottom and sides of two 9-inch round cake pans with oil. Punch dough to deflate; turn out onto a lightly floured work surface. Using a bench scraper or knife, divide dough into 4 equal portions, then divide each portion into 6 equal pieces. Form each piece of dough into a ball, pinching sides down and pinching at bottom to make smooth. Arrange 8 dough balls evenly around outer edges of each pan, and 4 in middle. Cover with clean kitchen towels and let stand until doubled in size, about 45 minutes.
Preheat oven to 350 degrees. Bake rolls until puffed and golden on top, about 20 minutes. Remove from oven and immediately brush with some melted butter. Let cool in pans at least 15 minutes before removing. Serve warm or at room temperature, brushed with more butter.
These rolls are best when served the same day they are made.