Make the Tangzhong: In a small saucepan, whisk flour, milk, and water over medium heat until thick and smooth. Let it cool slightly.
Prepare the Dough: Combine tangzhong, milk, cream, sugar, yeast, salt, and egg in a bowl. Gradually add flour and knead for 10 minutes. Add butter and knead until smooth and elastic.
First Rise: Cover and let the dough rise in a warm spot for about 1 hour or until doubled in size.
Shape the Rolls: Divide the dough into 9–12 portions, shape into smooth balls, and place in a greased baking pan.
Second Rise: Cover and let rise for another 45–60 minutes, until puffy.
Bake: Brush with egg wash and bake at 350°F for 18–22 minutes until golden brown.
Finish: Brush warm rolls with melted butter and cool slightly before serving.