In a small bowl, combine the warm milk, 1 tsp of sugar, and yeast. Stir and set aside for a few minutes until the yeast is frothy and active
In a large bowl, mix together the bread flour, remaining sugar, and salt. Add the sourdough discard, egg, and milk and yeast mixture, and tangzhong. Using a stand mixer with a dough hook attach, mix on low speed until just combined
While the stand mixer is running on low, add the softened butter piece by piece, until fully incorporated. Knead on medium - high speed for 10 - 15 minutes until the dough is smooth and passes the window pane test Shape the dough into a ball and place in a greased bowl. Cover and let rise until doubled in size, about 1 hour
Grease or line a 9in x 9in baking tin. Punch down the dough and turn out onto a floured countertop. Divide into 16 equal pieces (about 43g each, if you want to be very precise). Shape each piece into a roll by gently pinching together the ends of the dough and rolling on the countertop into a ball. Place the rolls in the baking tin and cover. Let rise until puffy and doubled in size, about 45 minutes
Towards the end of the second proof, preheat the oven to 350F. Brush the tops of the rolls with milk, then place in the preheated oven on the middle rack and bake for 18 minutes until the tops are golden brown and make a hollow noise when tapped