First, make the tangzhong. In a small saucepan, combine the milk and flour for the tangzhong. Cook over medium-low heat, whisking constantly, until thickened to a paste, 2 to 3 minutes. Immediately scrape the paste into a small bowl and cool on a wire rack for at least 5 to 10 minutes while you prep the dough.
Activate the yeast for the dough. In a small bowl, whisk together the milk, yeast, and a pinch of the sugar. Set aside for 5 minutes, or until bubbly and foamy, while you prep the other ingredients.
Make the dough. In the bowl of a stand mixer fitted with the dough hook, combine the remaining sugar, flour, salt, and egg for the dough. Mix with a rubber spatula until just combined and a slightly shaggy dough forms.With the mixer on low, add the tangzhong and milk and mix on low until completely shaggy. Add the butter one piece at a time, adding the next piece only after the previous one has been fully incorporated, scraping down the bottom and sides of the bowl as necessary. Once all the butter is combined, increase the mixer to medium-high and knead the dough for 10 minutes, or until tacky and slightly sticky. Proof the dough overnight. Tip the dough out onto a lightly floured counter. Dust your hands with flour and knead the dough once or twice into a rough ball. Scrape down the bottom and sides of the bowl to remove any excess dough. Spray the bowl with cooking spray and place the dough back in the bowl. Cover with plastic wrap. Refrigerate overnight.
Shape the dough and proof again. Once the dough has doubled overnight, lightly spray an 8-inch square pan with cooking spray. Uncover the dough and discard the plastic wrap. Punch down the dough and tip it onto a floured counter.Use a bench scraper to divide the dough into 12 equal pieces, about 2 ounces (57 grams) each. Shape each piece into a smooth ball. Place in the prepared baking pan, arranging the balls in three rows of four. Loosely cover with plastic wrap and let rise for 1 hour at room temperature. Meanwhile, prep the oven. About 30 minutes into the second rise, position a rack in the center of the oven and preheat the oven to 350°F.
Make the egg wash. In a small bowl, whisk together the remaining egg and milk for the egg wash. Use a pastry brush to brush the top and sides of each roll with the egg wash. Garnish the top of each roll with flaky salt. Bake the rolls. Bake for 28 to 30 minutes, or until the tops of each roll are golden brown. An instant thermometer inserted into the center of roll should read 200°F. Serve and store. Serve immediately. The rolls can be stored in an airtight container or zip-top bag at room temperature for up to 2 days.