Whisk 3/4 cup of whole milk into the hot tangzhong until completely smooth and cooled to room temperature or a little above.
Add the egg, vanilla extract, and sugar, whisking until smooth again.
In a separate bowl, combine self-rising flour, buttermilk powder, dough enhancer (if using), and yeast. Toss to combine thoroughly.
Stir 1/3 of the flour mixture into the wet ingredients. Add 1 tbsp. of the softened butter and begin kneading it into the dough. Alternate the remaining 2/3 of the flour mixture with the remaining 2 tbsp. of softened butter, kneading after each addition.
Continue kneading until the dough is smooth, elastic, and slightly tacky (but not sticky), about 8-10 minutes by hand or 5-7 minutes in a stand mixer.
Transfer dough into a bowl rubbed with butter or oil; flip the dough over a time or two so that the top surface is oiled as well as the bottom.
Cover the dough and let rise until doubled, about an hour.