Tangzhong Starter:

In a small saucepan, add in 3 tbsp of bread flour, and 1/4 cup of water, and whisk till the flour has dissolved and there are no more lumps. And in 1/4 cup of whole milk. Continue to whisk together to combine.

Set over low heat, stirring constantly with a spoon, for 3 to 5 minutes.

After 3 to 5 minutes of whisking over low heat, it should look like a thick paste but still pourable. It's done when the spoon starts to leave tracks on the bottom of the pot when stirring.

Turn off the heat and transfer the tangzhong starter into a small heatproof bowl, cover with plastic wrap touching the tangzhong to prevent a skin from forming, and let it come to room temperature. The starter will continue to firm up as it cools.

Dough:

Take 1 large egg and 4 tbsp of unsalted butter, and leave them on the counter to come to room temperature. Cut the butter into 4 pieces.

In a microwavable bowl, microwave 1/2 cup of whole milk till it is scalding, about 1 1/2 min in the microwave, to a minimum of 180 F (80 C). This deactivates the whey proteins in the milk, which may hinder gluten development. While it's hot, whisk 1 tbsp of granulated sugar into the milk.

Let the milk cool down by whisking it constantly, till it reaches 105 to 115 F (40-46C). When the milk is 105-115 F, whisk in 1 envelope (which is 1/4 oz or 2 1/4 tsp) of active dry yeast till it gets dissolved and let it proof for about 10 minutes. It should start to look foamy on top. If it does not get foamy, it means your yeast is not working. Start again with new yeast.

In the bowl of your stand-mixer, add in 2 1/2 cups of all-purpose flour, 2 tbsp nonfat powdered milk, 3 tbsp of granulated sugar, and 1 tsp salt. Using the regular beater attachment, mix this on medium-low speed to combine the dry ingredients. Stop the mixer, and add in the 1 large room temperature egg, the yeast-milk mixture, and the tangzhong mixture. Mix this together on low speed just until the ingredients are combined. With a spatula, scrape the batter off of the beater, remove the beater attachment, scrape the sides and bottom of the bowl to make sure there are no dry pockets of flour, and cover loosely with plastic wrap. Let the mixture rest for 5 minutes - this will allow the flour to soak up the liquid.

Attach the dough hook attachment, and knead on low speed (level 3) for 5 minutes.

After 5 minutes of kneading on level 3, add in the pieces of room temperature unsalted butter (4 tbsp total) one piece at a time, waiting for the butter to get incorporated before adding in the next piece, about 30 seconds to 1 minute between each addition of butter.

Once all the butter has been added, turn the speed up to medium (level 5), and continue kneading the dough for another 5 minutes, till the dough is smooth, springy, and no longer sticks to the side of the bowl.

Take a large bowl, oil it very well, and transfer the resulting dough into the oiled bowl. Cover the dough with plastic wrap, and let it sit at room temperature for 1 1/2 to 2 hours. This will allow it to rise and puff up (1st proof).

Ube Filling:

In a mixing bowl, add in 1/2 cup of ube halaya, 1/2 tsp of ube extract, and 1/8 tsp of salt. Mix this together until everything is well combined.

Assembling the Braided Ube Buns:

Prep a 12-cup muffin pan by buttering the wells very generously.

Grease a clean flat surface and rolling pin generously with vegetable oil. Punch down the dough, and turn it onto the greased surface. Shape the dough into an approximate rectangle shape by patting it down with your hands and squaring off the edges with a bench scraper. Starting with the dough that's already shaped into a rectangle makes it easier to get a more even rectangle when you roll it out. Roll your dough out into a rectangle that's 12" by 18".

Spread the 1/2 cup of prepared ube filling evenly over the entire surface of the dough using an offset spatula. Spread it all the way to the edges. Lift one of the long 18" edges toward the other long 18" end so that the dough is folded into half with ube filling in the middle.

You should now have a long rectangle of dough with ube halaya in the middle. Use a bench scraper and rolling pin as needed to reshape the dough so that it measures 6" by 18". Mark points 1.5" apart along the 18" side, then cut the dough to create 12 thin strips that are 1.5" by 6".

Taking one of these 1.5" by 6" strips at a time, cut the strip lengthwise into three strands of dough, leaving the edge that got folded intact. Braid the strips of dough together into one braid. Starting from the end that's intact, roll the braid up into a ball, and place it into one of the buttered wells of your muffin pan, seam-side down. Shape it with your hands as necessary to create a rounded and even shape. Cover loosely with plastic wrap to prevent the dough from drying out.

Repeat this with all of the 1.5" by 6" strips to fill all 12 wells of the muffin pan. When all of the braided buns are formed in the muffin pan and loosely covered with plastic wrap, let them rise again at room temperature until they are doubled in size, about 1 to 2 hours (2nd proof).

Preheat your oven to 350 F (175 C) about 15 minutes before the end of the 2nd proof. In a small bowl, prepare an egg wash by whisking 1 egg and 1 tbsp of water till they are well combined.

Brush the top of the risen buns with egg wash. Be careful not to drip too much egg wash down the sides of the buns, or brush too much of the exposed ube filling onto the dough.

Bake in the preheated oven at 350 F (175 C) for 15 to 17 minutes, or until the tops are golden brown. If you have a thermometer, buns are done when they read 185 - 190 degrees F (85 - 88 C) in the middle of each bun. When the inside goes past 200 degrees F (93 C), the buns will start becoming a little dry.

Once out of the oven, let them cool slightly, about 5 minutes. After 5 minutes, use a small knife to unstick the sides as needed and transfer them onto a cooling rack. Enjoy!

Store leftovers in an airtight container at room temperature, and will keep up to 5 days. Cooked buns can be frozen and will keep for up to 2 months.