Make the tangzhong by mixing flour, milk, and honey in a saucepan over medium-low heat.
Stir constantly until thickened (about 2-3 minutes). Transfer to a stand mixer bowl.
Add cold milk, egg, flour, yeast, salt, and softened butter to the tangzhong.
Knead with a dough hook for 10 minutes until the dough is smooth and pulls away from the sides.
Shape the dough into a ball, cover with plastic wrap, and let rise for 1 1/2 hours until doubled in size.*For overnight rise: refrigerate up to 24 hours after kneading, then proceed with the recipe. Grease a 9×13-inch pan and line with parchment paper.
Divide the dough into 15 pieces (about 82g each).
Shape each into a ball by folding the edges up and over, then rolling into smooth rounds. Place in the prepared pan.
Cover the pan and let rise for 45 minutes to 1 hour until the rolls are touching.
Preheat oven to 350°F. Brush the rolls with an egg wash (egg whisked with water).
Bake for 30 minutes until golden brown and the internal temperature is 190°F.
Melt butter and honey together. Brush over the rolls and sprinkle with flaky salt. Serve warm.