Add the tangzhong mixture, flour, milk, sugar, salt, and one egg to the yeast mixture. Mix on low speed for approximately five minutes, or until the dough begins to come together. Add flour, one tablespoon at a time, if the dough is too wet. Add a little more milk if it's too dry.
Add a spoonful of butter while the mixer is still running on low. Stir until completely combined. One spoonful of butter at a time, repeat with the remaining butter.
Place the balled-up dough in a basin that has been oiled. Cover with plastic wrap and let rise in a warm place until doubled, about 1 hour. There will still be some stickiness to the dough.
Apply nonstick cooking spray to a 9-inch springform pan.
When you're ready, punch the dough, transfer it to a floured board, and cut it into eight equal pieces, each weighing roughly 3 ounces.
Roll the ball beneath your palm in a circular motion while applying a light but firm and even pressure to each portion of dough, one at a time. To keep the dough in the middle of your palm, cup the sides of your hands.It could take some time to work the dough, but ultimately it will come together and the majority of the folds and lines should have been absorbed, creating a smooth ball of dough. Place the rolls, one at a time, in the prepared pan.
Cover and let rise again for 30 to 45 minutes, or until doubled.
Turn the oven on to 350°F.
Add the remaining egg and water to a small bowl and whisk quickly until creamy and blended.
Brush the egg wash over the tops of the rolls using a pastry brush.
Bake for 20 to 25 minutes, or until golden brown on top. (The baked rolls will have an internal temperature of 190°F.)
Make the garlic butter in the final minutes of baking the rolls.