In a small saucepan, whisk together the 3 tablespoons flour, 1/2 cup milk, and 1/4 cup water until smooth. Cook over medium heat, stirring constantly, until it thickens into a soft pudding-like paste and you can see the whisk leave tracks on the bottom, about 2 to 3 minutes. Scrape it into a small bowl and let it cool for 5 minutes.

In a large bowl or a stand mixer fitted with the dough hook, whisk together the 3 cups flour, yeast, sugar, and salt. Add the warm milk, egg, softened butter, and the cooled tangzhong.

Mix until a shaggy dough forms, then knead until smooth, soft, and just slightly tacky, about 8 to 10 minutes by hand or 6 minutes on medium-low with the dough hook. If it is very sticky, add flour a tablespoon at a time, but keep it on the soft side.

Place the dough in a lightly greased bowl, cover, and let it rise in a warm spot until doubled, about 60 to 90 minutes.

Punch down the dough and divide it into 12 equal pieces. Roll each into a tight ball and arrange them in a buttered 9x13 pan or a 10-inch cast iron skillet.

Cover and let the rolls rise again until puffy and touching, about 30 to 45 minutes. Near the end, heat the oven to 375F.

Bake for 18 to 22 minutes, until the tops are deep golden brown. Brush with the melted butter and sprinkle with flaky salt. Cool for 5 minutes, then pull apart and serve warm.