Heat the milk until tiny bubbles form around the edges and the milk is steaming. Pour into the bowl of a stand mixer fitted with the dough hook.
Add the water, butter, and sugar. Let the mixture cool until warm (if you dip a finger in, it should feel like a warm bath - not too hot, not too cold, just right).
Stir in the yeast and let rest for 5-7 minutes until the yeast is foamy (technically, you don't need to let the yeast activate before adding the flour, but I find in yeast bread recipes with milk, the dough rises quicker/better if I let the yeast activate first).
Add the egg, salt and flour (start with 3 3/4 to 4 cups flour) and mix, adding additional flour gradually, until the dough clears the sides of the bowl and forms a soft, sticky ball. Don't over flour the dough.
Knead for 5-6 minutes until the dough is soft and smooth.
Let the dough rise in a warm spot until doubled, about an hour or so. (I usually let the dough rise right in the mixing bowl, but you can transfer to a lightly greased bowl and cover.)
Grease a 9X13-inch pan. Lightly punch down the dough and turn out onto a lightly greased countertop.
Divide the dough into 12 equal pieces (3 to 3.5 ounces each). Roll each piece into a round ball and place in the prepared pan, spacing 1/2-inch or so apart (four rows of three rolls).
Preheat the oven to 375 degrees F. Place an oven rack in the middle position.
Cover and let the rolls rise until doubled. The rolls should be very puffy with the sides touching.
Bake the rolls for 15-20 minutes until golden on the tops and bottoms. Butter the tops of the rolls immediately out of the oven.
Notes
Sugar: if you want a sweeter roll, increase the sugar by 1-2 tablespoons.Flour: I haven't tried these rolls with all whole wheat flour, but they work well subbing in 1-2 cups of the all-purpose flour with whole wheat flour (preferable white whole wheat, kamut or einkhorn for a lighter texture).Make-Ahead: the rolls can be placed in the baking pan, covered with greased plastic wrap, and immediately refrigerated (don't let them rise in the pan first). They can rest in the refrigerator for 24 hours (they'll rise some in the fridge). Take them out of the refrigerator in time to come to room temperature and rise fully if they haven't already. Bake as directed in the recipe. Baked and cooled rolls can be frozen in resealable bags.