To prepare the tangzhong (water roux), add 1 oz (30 g) of bread flour to 5 oz (150 g) of water in a saucepan. Stir to combine.
Bread Starter: Turn the heat to low and stir the mixture until it becomes paste-like. Use a thermometer to monitor the temperature and remove it from heat once it reaches 149°F (65°C). Transfer to a glass or stainless steel bowl, cover with plastic wrap, and let it cool to room temperature or refrigerate for later use.
Tangzhong Dough: Attach the dough hook to your mixer. Add the wet ingredients to the bottom of the mixing bowl, then add the salt and sugar, followed by the dry ingredients, with the yeast added last. Set the mixer to low speed. Once the ingredients are well combined and the dough isn’t sticking to the sides, increase the speed to medium until gluten forms. This may take about 5-6 minutes, depending on your mixer.
Slowly add the butter to the dough in 3 batches and let it mix at a slow speed until the dough no longer appears sticky. Be patient; this step may take some time. Once all the butter is kneaded into the dough, set the speed to medium and knead for 6-8 minutes until passing the windowpane test.
Grease a bowl and shape the dough into a round. Place it gently in the bowl, cover with plastic wrap, and let it rise for 40 minutes or until it doubles in size. Dust your finger with flour and insert it into the center of the dough. If the dough doesn’t rise back, it’s ready for the next step.
Divide the dough into 9 portions and shape each into a ball. Cover with plastic wrap and let rest for 10 minutes. Line the baking trays with parchment paper. Next, flatten each dough ball to release air, then turn it bottom side up and roll it tightly from one end to the other, sealing the opening. Roll each piece into a log about 8-10 inches long and place them on the baking tray, spaced about 2 inches apart. Cover the tray with plastic wrap and let them proof for another 40 minutes, or until nearly doubled in size.
Preheat the oven to 350°F (176°C). Prepare the egg wash and brush it on the dough. Sprinkle with cheese and chopped parsley. Bake for 14-16 minutes, then transfer to a cooling rack. For freshness, store the cheese breadsticks in an airtight container.