Melt the Butter (3 tablespoon).
To prepare the tangzhong roux: mix the Bread Flour (50 gram) with Milk (125 milliliter) and Water (125 milliliter) until all flour dissolve and has no lumps.
Cook in low-medium heat and keep stirring, if you have a thermometer, cook the water roux up to 65 degrees C (150 degrees F). This amount should be enough to make 2 batches; you can leave it in the fridge for up to 3 days.
Transfer the tangzhong roux to a bowl to cool down, cover the water roux with cling wrap to prevent the surface from getting dried out.
Put Milk (110 milliliter), tangzhong roux (100 grams), Brown Sugar (45 gram), Salt (1 teaspoon), Bread Flour (350 gram), and Instant Dry Yeast (1 1/2 teaspoon) into bread maker according to instruction.
Select Dough mode in your bread machine. Once you see everything start to form a dough, pour in the melted Butter (40 gram) and Dried Cranberries (1/2 cup).
Once your bread maker finishes the dough process, poke a hole into the dough. If it doesn't bounce back, your dough is finished.
Transfer the dough onto a slightly floured bench top, press the dough down to get the air out. Cut into 9 pieces. Use a rolling pin, roll each piece flat and fold it like a dumpling to make a ball like dough.
Line a square 18cmx18cm pan with baking paper and transfer the dough balls into the pan.
Cover the pan with a towel or cling wrap to prevent the dough drying out. Let the dough rise for another 1-1.5 hours until it rises enough that the dough balls are touching each other.
About 10 minutes before the dough is finished rising. Preheat the oven to 170 degrees C (340 degrees F).
Combine the sugar, flour and water until well combined. Transfer to pipping bag.
Brush the bun with Egg (1) on the top and pipe the flour mixture on top to create the “Cross”.
Bake for 20-23 minutes. Once finished let it cool down for 5 minutes on a wire rack.