Mix together warm milk, yeast, and sugar and set aside for 5 minutes. The milk should feel comfortably warm to the touch, but not hot, otherwise it will kill your yeast. You should notice the yeast beginning to foam. If there is no sign of activity, the yeast may be dead. Try again with a fresh batch.
Add flour, egg, tangzhong, and salt to the milk and yeast mixture. Using a stand mixer fitted with a dough hook attachment, mix on low speed until a shaggy dough forms. Slowly add in pieces of softened butter while the mixer runs on medium-low speed. Continue mixing for 15 - 20 minutes until the dough is smooth and passes the window pane test. Please see notes if you would like to make by hand.
Shape into a ball and place into a lightly greased mixing bowl. Cover with plastic wrap and let rise for 40 – 60 minutes until roughly doubled in size.
Grease a baking tin and set aside. Turn the dough onto a lightly floured surface. Knead a few times to remove air bubbles.
Roll the dough into a 10 x 15 inch rectangle. Spread the surface of the dough with about 1 tbsp of vegetable shortening. Sprinkle with about 1/2 of the chopped scallions. Roll the dough up so that you have a 15 inch tube. Then, coil the tube so that you end up with a round piece of dough that is swirled like a snail shell.
Repeat the layering process - roll out this piece of dough into a 10 x 15 inch rectangle. Spread the surface of the dough with about 1 tbsp of vegetable shortening. Sprinkle with 1/4 tsp of salt and the remaining chopped scallions, leaving a few scallions for garnishing the top . Roll the dough up so that you have a 15 inch tube. Using a knife or a bench scraper, cut the tube in half length-wise - the knife should be parallel to the tube. Turn the inside of the tube up and twist the two pieces over each other until you have a braid. Tuck the ends of the braid underneath and place the dough into your prepared baking tin. This twisting method is similar to making a babka.
Preheat oven to 350 degrees Fahrenheit. Allow the loaf to proof for 45 – 60 minutes. When you press your finger into the loaf, it should slowly come back about halfway. Brush the top of the loaf with milk and sprinkle with sesame seeds and leftover scallions.
Place in the lower third of the oven and bake for 35 minutes until golden brown and makes a hollow sound when tapped.
Remove from oven and let cool for 10 - 15 minutes in the tin. Remove from tin and place on a wire cooling rack to finish cooling.