In the bowl of a stand mixer fitted with the dough hook, combine the cooled tangzhong, yeast mixture, melted butter, vanilla, eggs, salt, cinnamon, nutmeg, shredded carrots, and both flours. Mix on low speed until a soft dough forms.
1/4 cup unsalted butter, 1 teaspoon vanilla extract, 2 large eggs, 1 1/4 teaspoons salt, 1 teaspoon ground cinnamon, 1/4 teaspoon ground nutmeg, 2 1/2 cups all-purpose flour, 1 cup bread flour, 1/2 cup finely shredded carrot