Whisk ⅔ cup whole milk and 3 Tbsp. (23 g) all-purpose flour in a small saucepan to combine. Place over medium heat and cook, whisking constantly, until thickened and pasty, 2–3 minutes. Transfer mixture (this is your tangzhong) to the bowl of a stand mixer and attach dough hook. Add ¼ cup (50 g) sugar, one ¼-oz. envelope instant yeast (about 2¼ tsp.), 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt, 2 large eggs, ¾ cup whole milk, and remaining 4 cups (500 g) all-purpose flour and mix on low speed until a shaggy dough forms, about 1 minute. Add ½ cup (1 stick) room-temperature butter, cut into pieces, a piece at a time, beating to incorporate after each addition before adding more. When all of the butter has been added, increase speed to medium-high and beat until dough is smooth and supple, about 10 minutes.
Transfer dough to a large bowl lightly coated with vegetable oil; turn to coat. Cover with plastic wrap and let sit in a warm, dry spot until doubled in size, about 2 hours. (Or, place in refrigerator and chill until doubled in size, at least 8 hours and up to 12 hours).
Brush a 13×9″ baking dish well with some of 6 Tbsp. unsalted butter, melted, slightly cooled. Punch down dough a couple of times to release air. Portion equally into 24 balls (about 45 g each). Cup your hand over a ball and drag across surface, moving your hand in a circular motion, to create a smooth, tight ball. Turn over so smooth side is facing down. Press out to a 4×2″ rectangle and brush with melted butter. Fold in half and place in prepared baking dish. Repeat with remaining balls, making sure to arrange buns so folded edges are all facing the same direction. Cover with a kitchen towel; let sit until slightly puffed, 45–60 minutes.
Place a rack in lower third of oven and preheat to 350°. Beat remaining 1 large egg and 1 Tbsp. whole milk in a small bowl to combine. Brush over buns, then sprinkle with flaky sea salt. Bake, rotating front to back halfway through, until golden brown and tops are crisp to the touch, 35–38 minutes.