1-Line a 9×13 baking pan with parchment: Line a 9×13 baking pan with parchment paper. Set aside.
2-In the small pot, whisk together the: In the small pot, whisk together the water and flour. Cook over medium heat, whisking constantly, for 2-3 minutes until the mixture is a paste-like slurry. Scrape it into a bowl and set it aside.
3-In a stand mixer bowl, mix the: In a stand mixer bowl, mix the flour, sugar, yeast, and salt. Add the milk, eggs, and tangzhong to the bowl. Knead on low speed with the dough hook for 2 minutes, until the dough forms a rough ball.
4-With the mixer running on low, add: With the mixer running on low, add the softened butter one tablespoon at a time, letting it fully incorporate before adding more. Then, continue kneading for an additional 12 minutes, until the dough appears smooth and elastic and feels tacky. *If the dough is overly sticky, you may need to add an extra tablespoon of flour.*
5-Portion the dough into 15 pieces, each: Portion the dough into 15 pieces, each ~83 grams. They are fine to leave on your counter, but I would loosely cover them with plastic wrap to prevent the dough from drying out while you are shaping them.
6-Shape each piece into a rough ball,: Shape each piece into a rough ball, then roll it into a ~5×6-inch rectangle. Fold one long side toward the center, then fold the other long side over it. Roll it up from the short end into a log, then pinch the seam to seal. Place it in the baking pan and repeat with the remaining dough. Do not use flour on your work surface.
7-If you prefer a more traditional round: If you prefer a more traditional round dinner roll: Use a “claw” hand to roll each piece on an unfloured surface into a tight, taught ball, then pinch the bottom seam to seal it together. Place into the prepared baking pan.
8-Cover the pan with plastic wrap and: Cover the pan with plastic wrap and set aside to double in size, about 1.5-2 hours. You’ll know they are ready when you gently touch the sides of the rolls, and the dough feels light and airy. Meanwhile, preheat the oven to 350°F (180°C).
9-Before baking, brush the tops of the: Before baking, brush the tops of the rolls with the beaten egg.
10-Bake for 30-40 minutes, or until the: Bake for 30-40 minutes, or until the tops are deeply golden brown.
11-Once baked, remove the pan from the: Once baked, remove the pan from the oven and place it on a wire rack. Immediately brush the tops with melted butter and sprinkle with flaky sea salt if desired. Serve warm and enjoy!