In a small saucepan set over low heat, whisk together the whole milk, water, and flour until smooth. Cook until the mixture thickens, about 3 to 5 minutes. Remove from the heat and set aside.
To Make the Dough
In the bowl of a stand mixer, combine the flour, dry milk, sugar, salt, and yeast. Mix on low until combined. In a 2-cup glass measuring cup, warm the milk in the microwave. Whisk in the starter, eggs, and melted butter. With the mixing speed on low, gradually add the wet ingredients. Mix just until a dough forms. Switch to the dough hook and knead for 15-20 minutes.
Shape the dough into a ball and transfer it to a lightly buttered bowl. Cover with plastic wrap and place it into a warm oven (turn it on to 200 degrees Fahrenheit/93 degrees Celsius, then turn it OFF when it reaches that temperature) for 60 to 90 minutes or until doubled.
Gently deflate the dough by pressing a fist into the center. Divide into 12 equal pieces and shape each piece into a ball. I like to pinch the bottom of the dough balls and then form a C shape with my hand, rolling the ball on the counter to make a smooth, taut circle. That way, the rolls have a more uniform shape. If you have a kitchen scale, weighing it will help with that too.
Place the rolls into a lightly buttered quarter sheet pan or 9x13-inch pan. Cover with a clean kitchen towel and return to the oven. Let the rolls rise for 40 to 50 minutes, or until puffy.
Preheat the oven to 350 degrees Fahrenheit (177 degrees Celsius). In a small bowl, whisk together the egg and milk. Brush the tops of the rolls and bake for 20 to 25 minutes, or until golden brown on top (a digital thermometer inserted into the center of the roll should reach 190 degrees Fahrenheit/88 degrees Celsius).
Remove the rolls from the oven and brush the tops with butter. Allow the rolls to cool for 10 minutes before pulling apart and serving.
Notes
-If you don't have an electric mixer, try kneading the dough for as long as you can manage. The dough can also be kneaded in a Bosch mixer or bread machine.-Freeze these rolls and let them thaw at room temperature for a few hours before warming.To STORE your Dinner Rolls, keep them in an airtight container at room temperature for up to 3 days. For longer storage, FREEZE them by wrapping them tightly in plastic wrap and placing them in a freezer-safe bag, where they’ll stay fresh for up to 3 months.To REHEAT, warm them in a 350°F (175°C) oven for about 10 minutes or microwave for 20-30 seconds.