In a small saucepan over medium-low heat, whisk together the flour and milk. Stir constantly for about 2 minutes until it thickens into a smooth paste. It should resemble a soft pudding.Remove from heat and let it cool for about 5 minutes before using. In the bowl of a stand mixer, whisk together the warm milk and sugar. Sprinkle the yeast over the top and let it sit for 5–10 minutes until it becomes foamy.
Add the cooled tangzhong, egg, flour, and salt into the yeast mixture. Using a dough hook, mix on low speed until a shaggy dough begins to form.
With the mixer running, add the softened butter one tablespoon at a time.Increase to medium speed and knead for 8–10 minutes. Transfer the dough to a lightly greased bowl. Cover and let it rise in a warm, draft-free space for 1 to 1.5 hours, or until doubled in size.
Gently punch down the dough to release air. Divide it into 12 equal portions (about 60g each).Roll each piece into a smooth ball by tucking the edges underneath and rotating it against your work surface. Place the dough balls into a greased 9x13-inch baking dish. Cover the shaped rolls and let them rise again for about 45 minutes.
Preheat your oven to 350°F (175°C). Brush the tops lightly with melted butter and bake for 20–25 minutes, or until golden brown on top.
While the rolls bake, mix together the melted butter, minced garlic, and parsley.As soon as the rolls come out of the oven, brush them generously with the garlic butter. Sprinkle with freshly grated Parmesan and a pinch of flaky salt.
