In a small bowl, whisk the warm milk and sugar until dissolved. Sprinkle the yeast over the top and let sit 5 minutes, until foamy.
3/4 cup whole milk, 2 1/4 teaspoons active dry or instant yeast, 1/4 cup granulated sugar
In the bowl of a stand mixer fitted with the dough hook, combine the cooled tangzhong, yeast mixture, melted butter, eggs, vanilla, salt, and both flours. Mix on low speed until a soft dough forms.
1/4 cup unsalted butter, 2 large eggs, 1 teaspoon vanilla extract, 1 1/4 teaspoons salt, 2 1/2 cups all-purpose flour, 1 cup bread flour
Knead on medium speed for 5 to 8 minutes, until smooth and elastic. Gradually add the softened butter one tablespoon at a time until fully incorporated. The dough will be soft and slightly sticky.
5 tablespoons unsalted butter
Transfer to a greased bowl, cover, and let rise in a warm place for 1 to 2 hours, until doubled.
For easier shaping, refrigerate the dough for 2 to 8 hours or overnight after the first rise.