In a stand mixer with the hook attachment, mix the flour, sugar, yeast, milk powder and salt until combined.
Add warmed milk, tangzhong and egg and kneed on low until ingredients are combined for 5 minutes.
Once the dough has formed, add the softened butter. Increase speed and kneed until combined around 5-7 minutes. The dough should be tacky but not sticky.
Divide the dough into two equal parts. Set aside one half of the doughs.
Place the second dough back in the mixer, add the ube paste and knead until combined.
Proof the Doughs. Shape the doughs and place in two lightly greased bowls. Proof in a warm place for 1-2 hours until doubled.
Shape the Dough. When the dough has doubled, remove from the bowl. On a lightly floured surface, roll out the white dough into an 8.5 x 12 inch rectangle. The dough should be pretty thin. Then roll out the ube dough into the same size of the white dough. Place the ube dough on top of the white dough and roll out the dough to have the doughs merge. Carefully roll the dough down the longer 12 inch side of the dough. Pinch at the end to seal the seam.
Second Proof Dough. Place the logs seam side down in a 8.5 x 4.5 loaf pan. Next, proof in a warm place for another 30 min to 1 hour until doubled in size.
Bake. Preheat oven to 355 degrees. Brush with egg wash and bake for 30 minutes.