In a small saucepan, combine the whole milk, flour, and water. Set the pan over medium-high heat and whisk constantly until the mixture begins to thicken into a smooth, paste-like consistency. This will take a few minutes.
1/2 cup Whole Milk, 1/2 cup Water, 1/4 cup Bread Flour
As soon as it begins to thicken, remove from heat and continue whisking for another 30 seconds. If you leave it on the heat, it will become too dense and you will have to start over.
Set aside and let the tangzhong cool to room temperature.
Activate the Yeast
In the bowl of a stand mixer, mix the yeast with warm water and sugar, stirring gently. Let the mixture sit for about 10 minutes, or until it becomes foamy.
Add the flour, sugar, salt, milk, Tangzhong starter, and eggs to the yeast mixture.
4 3/4 cups Bread Flour, 2 teaspoons Salt, 1/3 cup Sugar, 2 large Eggs, 1/2 cup Whole Milk
Mix the Dough
Using the dough hook attachment, mix on low speed until the dough comes together and looks shaggy. When the dough begins to come together, increase the speed to medium.
Add the Butter
Gradually add in the melted butter. Continue kneading for about 5-7 minutes, or until the dough is smooth, elastic, and pulls cleanly away from the sides of the bowl. The dough should feel slightly sticky but not overly wet.
6 Tablespoons Butter
Shape and Rise
Transfer the dough to a lightly floured surface and shape it into a smooth ball. Place the dough ball into a lightly greased bowl, cover it with a damp towel, and let it rise in a warm area for 1-2 hours, or until it has doubled in size.
Divide and Shape Rolls
Once the dough has risen, divide the dough into about 16 rolls.
Roll each piece into a tight, smooth ball by tucking the edges under and rolling with your palm on the work surface. To shape, roll each portion into a ball by cupping your hand over it and rolling in a circular motion on the counter.
Second Proofing
Line a 9x13 inch sheet pan with parchment paper and arrange the dough balls. Cover with a damp towel and allow to rise again for another 20 minutes while your oven is coming to temperature.
Apply the Egg Wash
In a small bowl, beat together one egg with a splash of whole milk to create an egg wash. Gently brush the tops of the rolls with the egg wash to encourage a glossy, golden-brown finish.
1 large Egg, 1 tbsp whole milk
Bake the Rolls
Preheat your oven to 350°F. Bake the rolls for 22-28 minutes on a middle rack. To prevent over-browning, you can tent a piece of foil over the rolls halfway through baking if needed.
Finish with Butter and Salt
Remove the rolls from the oven and, while still warm, brush them with melted butter and sprinkle with flaky salt.