Heat your whole milk in the microwave for 30 seconds, then whisk in your instant yeast and let sit for 10 minutes.
To the bowl of your stand mixer, add in your flour, salt, and sugar and combine. Using the dough hook attachment, add in your roux, the milk and yeast mixture, and your egg (quickly run your egg under some warm water to bring the temperature up slightly) and mix on low (scraping the sides if needed) until the dough starts to come together. Increase speed to medium low and add your butter 1 tablespoon at a time. Let mix for 7 minutes on medium-low once the butter is incorporated.
Lightly grease a large bowl with butter. Lightly flour your hands, remove the dough from the stand mixer, and form into a tight ball. Place the ball of dough into your buttered bowl (picture two), cover with a damp towel and let rise for 1 -2 hours, or until doubled in size (picture three).
Lightly grease your baking dish with butter, set aside. Lightly flour your work surface, and remove your dough from the bowl. Separate your dough into 9 equal pieces and roll each one into a ball, then place in your buttered baking dish. You want the dough to be touching or very close to touching (picture 4). Cover with a damp towel and let rise for another 1 - 2 hours.
Preheat your oven to 350 degrees Fahrenheit.
In a small bowl, whisk together one egg and about 2 teaspoons of water, and using a pastry brush, brush the egg wash lightly over the rolls. Then bake your rolls at 350 degrees Fahrenheit for 25 minutes. Let cool, and set aside.