In a bowl, mix active sourdough starter, flour, sugar, and cold milk. Cover and rise in a warm spot for 4 hours. You want the levain to be active but not overly sour and liquid.
Tangzhong
Mix flour and milk in a small pot. Heat up over medium heat. Stir until the mixture thickens and reaches a temperature of 150°F/65°C.
Transfer to a clean bowl. Cover with plastic wrap that is touching the entire surface of the tangzhong. Cool to room temperature.
Main Dough
Add all ingredients (except salt and butter) to the bowl of your stand mixer fitted with a dough hook. Mix until the dough comes together. Cover, and autolyse for 30 minutes.
Add the salt, and knead at medium speed for 10 minutes. At this point, the dough should be tacky but not sticking to the fingers.
Transfer the dough to your work surface and divide it into four equal pieces. Roll into seamless rounds, cover, and rest for 10 minutes.
With a rolling pin, roll each round into an oval shape about 8 in/20 cm long. Grab one of the short sides of the oval and roll it up like a roulade. Repeat on all four pieces - cover and rest for 10 minutes.
With the seam side up, use a rolling pin to flatten each dough piece into long ovals again. Shape the ovals into roulades as you did previously. Transfer to a greased 10-inch bread pan.
Cover the bread pan and proof in a warm spot, ideally at 78°F/25°C, for 8-12 hours or until the dough fills up about 85% of the tin.
Bake
Preheat your oven to 355°F/180°C.
Bake the milk bread for 30-35 minutes, until golden brown. Cover the surface with aluminum foil if it browns too quickly—the internal temperature when finished: 200°F/95°C.
Take the bread out of the bread tin and brush the surface with melted butter (optional). Transfer to a wire rack and cool completely before slicing.