Make the tangzhong. Combine the water, milk, and bread flour in a medium saucepot and heat over low heat. Cook the mixture, whisking constantly until thick paste forms, about 3 minutes. Remove from heat and let cool completely.
Combine the tangzhong with the remaining ingredients for the milk buns except for the cheddar (bread flour - salt) in the bowl of a stand mixer. Knead the dough with the mixer or by hand until a rough dough forms. Add ¾ cup shredded cheddar and continue to knead until the dough is soft and smooth, about 7-8 minutes. Transfer the dough to a clean bowl, cover, and let rise in a warm area for 1 hour or until doubled in size.
Meanwhile, cook the bacon either on the stovetop or in the oven until browned and crispy. If cooking in the oven, lay the bacon on a sheet pan lined with a wire rack and bake at 400 degrees F for 20 minutes or until crispy. Lay the cooked bacon on paper towels to drain the excess fat. Break the bacon into small pieces and set aside.
Punch down the dough. Divide the dough into 15 equal portions, about 54 grams each. Shape each portion into a round ball and arrange them in a 9x13 inch baking dish into 3 rows of 5 rolls. Loosely cover the rolls and let rise for 45 minutes.
Preheat oven to 375 degrees F.
Uncover the rolls and use a bread lame or scissors to cut a diagonal slit across the top of the buns, about 1 centimeter deep. Brush the rolls with egg wash.
Season the chopped scallions with 2 tablespoon melted butter and 1 teaspoon sugar and spoon the scallion topping into the slits. Sprinkle the crumbled bacon and ¼ cup shredded cheddar on the buns and bake for 20-25 minutes or until golden brown. Remove from heat and let cool for 10 minutes. Serve warm.