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In small saucepan, add milk, water, and flour and whisk to combine. Continue whisking over medium low heat until mixture starts to get very thick and paste-like. Remove from heat and set aside to cool slightly.
In the bowl of stand mixer, add flour, sugar, yeast, dry milk, and salt. Whisk to combine. With mixer running on low speed with the dough hook attached, add the butter and then slowly add milk. Add egg and mix briefly. Stop mixer and add tangzhong. Increase the speed to medium high and knead for 5 minutes.
Scrape dough together and cover bowl (or transfer to another bowl). Place in a warm, draft-free area and allow to proof until doubled, 1-2 hours. Refrigerate dough overnight (at least 8 hours).
When ready to make rolls, remove dough from refrigerator and dump onto a well-floured surface. Knead until smooth, adding additional flour as needed to create a smooth, less sticky dough.
Divide dough into portions. For dinner rolls, I divide into 2-ounce pieces. If making hamburger or hot dog buns, 3-4 ounces is standard. Flatten each piece slightly and pull in the dough from the sides to create a taut dough. Pinch the dough together and turn over to where the pinched dough (the seam) is on the bottom. Put ball of dough on a smooth surface that isn't floured (the flour will cause the dough to slide around and you want friction). Put your hand over the dough with the outer edge of your palm where your pinky finger is creating a "cage" over the roll (the side of your hand should always touch the surface of the counter). Move your hand in a counter clockwise direction, pushing the dough around inside the "cage" until the roll is nice and round. If it sticks a little, that's okay. Just use your bench scraper to lift it up and place it into a prepared pan*.
Cover rolls with plastic wrap and place in warm, draft-free area to rise until not quite doubled in size, 30 minutes to an hour. Before baking, brush rolls with melted butter. Bake in a preheated 350° oven for about 20 minutes, or until golden brown and they bounce back when lightly touched. Brush with remaining melted butter and, if desired, cover with a towel to further soften the crust.