Combine the butter and milk in small saucepan and heat until the butter has melted and the mixture is no hotter than 110°. Set the pan aside to cool.
Use an instant-read thermometer to make sure the warm water is 110°F and then combine the warm water, yeast and sugar together in a large bowl or the bowl of your electric stand mixer. Proof the yeast (let it sit for 10 to 20 minutes, until the mixture is foamy). If the yeast is active, it should foam in the bowl. If yeast does not foam up, discard and start again with new yeast.
Once the yeast has proofed, add the eggs, salt and cooled milk and butter mixture to the bowl and stir until combined. Gradually add 6 cups of the flour, stirring until a dough begins to form. Add more flour as needed - the dough should be soft and elastic, but not sticky. Knead the dough in the electric stand mixer for 5 minutes or by hand for 15 minutes.
Grease the inside of a large bowl with oil and transfer the dough to the bowl, turning it over a few times to make sure it is lightly but completely coated with oil. Cover the bowl with a clean damp towel and place it in a warm place to rise for 1 hour, until the dough has doubled in size.
Grease a half sheet pan (an 18-inch by 13-inch baking sheet for 24 portions) or a quarter sheet pan (a 9-inch by 13-inch baking sheet for 12 portions) lightly with butter.
Punch down the dough and divide it into four equal portions. Cut each portion into 6 even pieces so that you have 24 portions of dough. (If you want the rolls to be perfectly even, weigh the dough on a scale and then weigh each dough ball.) Roll each portion of dough into a smooth ball and place all the balls in evenly spaced rows on the buttered baking pan. (The dough balls can be frozen at this point if making ahead – see notes.)
Cover the sheet pan with a damp towel and let the dough balls rise for another hour, until they have doubled in size. The dough balls should be touching at this point.
Pre-heat the oven to 375°F while the dough balls are rising.
Bake the rolls for 20 to 25 minutes, until they are golden brown on top. As soon as you remove from the rolls from the oven, brush them with melted butter.
Line a serving bowl or basket with a clean tea towel and place the rolls inside, topping with another towel to keep them warm. Serve immediately.
Make ahead:
These rolls will stay soft for up to a week. Refrigerate them after two days.
To make 1 day ahead: Bake the rolls as instructed and then allow them to cool completely on a cooling rack. When they are cool, don’t separate the rolls, but instead wrap them well on the sheet pan with plastic wrap. Before serving, cover the sheet pan with foil and warm all the rolls together in a 350°F oven for 10 minutes. Re-heat them uncovered if you want crusty rolls.
To make up to 2 weeks ahead: Follow the recipe through step 6. Place the dough balls on the baking sheet. Grease one side of a piece of plastic wrap and wrap the rolls on the sheet thoroughly to make sure no air gets in. Freeze. Take the rolls out of the freezer 4 hours before you are ready to bake and let the dough balls come to room temperature. Then, let them rise according to step 7 until they have doubled in size and proceed with the recipe.