Step 1Sprinkle the yeast over the warm water in a small bowl and let stand until bubbly, about 5 minutes. In a 1-quart bowl, beat the egg well, then whisk in the sugar, salt, milk, 1/2 cup of the butter and the yeast and water. Gradually stir in the flour, adding if needed up to 1/2 cup more, just enough to make a soft, very sticky dough. Use a little of the remaining butter to grease a 2-quart bowl. Transfer the dough to the bowl. Cover tightly with plastic wrap and refrigerate at least 6 hours, or overnight.
Step 2Remove the dough from the refrigerator and let stand at room temperature for 1 1/2 hours.
Step 3Heat the oven to 400 degrees. Butter either a 12-cup muffin tin or 2 (8-inch) cake pans.
Step 4Knead and shape the dough into rolls. For cloverleaf, tear off tablespoon-sized balls and place 3 in each muffin cup; for dinner rolls, divide the dough into equal rounds and arrange half an inch apart in pans. If the dough is too sticky, butter your hands well when you handle it. Cover the rolls with a dish towel and let rise in a warm spot until doubled in bulk, 30 minutes.
Step 5Lightly brush the rolls with the remaining melted butter. Bake until the rolls are just browned on top and bottom, 15 to 18 minutes. Serve warm with plenty of butter.