Combine the milk and water and heat until the mixture is 110-115° F. (I usually combine them in a glass liquid measuring cup and warm in the microwave. You can also warm them in a pot on the stove.)
Pour the warm milk and water into the bowl of a stand mixer. Add the granulated sugar and sprinkle the yeast over the top. Whisk gently to combine. Let the mixture sit for 5-10 minutes, until the yeast starts to foam and/or bubble.
Add the salt, egg and all-purpose flour. Mix on low speed using the paddle attachment. Once combined, add the 4 tablespoons of softened butter and continue mixing until the butter is incorporated.
Switch to the dough hook and knead the dough on low speed until it is smooth and only slightly sticky to the touch, about 8 minutes. The dough is supposed to be slightly sticky, but if it is very sticky you can add a little more flour, 1 tablespoon at a time. Avoid adding too much flour, because this will make the rolls more dense instead of light and fluffy. The dough should stick to your finger a little bit when you touch it but not be so sticky that it won't be workable.
Transfer the dough to a large lightly oiled bowl and turn once to coat all sides of the dough ball with oil. Cover the bowl with a clean kitchen towel or plastic wrap and let rise in a warm place for 30 minutes to 1 hour, or until doubled in size.
Lightly grease a 13x9-inch rectangular baking dish with cooking spray or butter.
Transfer the dough to a lightly floured work surface and knead a few times by hand to release any air bubbles.
Cut the dough into 15 equal sized pieces. To do this, first cut the dough into 3 equal pieces. Then cut each of the 3 pieces into 5 pieces. The dough balls should be approximately the same size, but they do not need to be exactly the same. (If you have a kitchen scale you can divide the total weight of the dough by 15 for an approximation of how much each roll should weigh.)
To shape the rolls, first make each piece of dough into a ball by tucking the edges of the dough ball underneath itself. Then place the dough ball on a clean work surface and, cupping your hands around the ball, use your palms to spin it on the work surface until it becomes a tighter ball. Place the rolls into the prepared baking dish.
Cover the rolls in the dish with a kitchen towel or plastic wrap and let rise for 20 minutes.
Meanwhile, preheat the oven to 375° F.
Bake the rolls, uncovered, for 19-23 minutes, until the tops are golden brown and the center rolls are baked through.
Remove rolls from the oven. Brush the 1 tablespoon of melted butter on the tops of the warm rolls. Serve warm.