In a small saucepan, whisk together ¼ cup of water and ¼ cup of all-purpose flour. Cook over medium heat until thickened, about 1 minute. Remove from heat and stir in ½ cup of whole milk. Set aside to cool.

In a stand mixer, combine the cooled tangzhong with ¾ cup of cold milk, 2¼ teaspoons of active dry yeast, 2 tablespoons of granulated sugar, 1 teaspoon of fine salt, and 2 large eggs. Add 3 cups of flour and mix for about 10 minutes. Incorporate ¼ cup of softened butter and mix for 8 minutes until elastic.

Transfer dough to a greased bowl, cover, and let rise in a warm area until doubled in size, about 2 hours.

Punch down dough and divide into 12 pieces, shaping each into a smooth ball. Place in a greased 9x13-inch baking pan. Let rise until puffy, about 1½ hours.

Preheat oven to 350°F (175°C) with two racks inside.

Whisk together 1 egg and 1 tablespoon of milk. Brush this egg wash over the rolls.

Bake for 20 to 25 minutes until golden brown and hollow-sounding when tapped on the bottom. Let cool in the pan for a few minutes.

Mix ¼ cup of melted butter with 1 tablespoon of dried herbs, 1 teaspoon of garlic powder, and ¼ cup of Parmesan cheese. Brush over warm rolls and sprinkle with flaky sea salt. Serve warm.