Add the sourdough starter, milk, tangzhong, and sugar to the bowl of a stand mixer fitted with the dough hook. Mix briefly until combined. All the levain, 240 g milk, All of the cooled tangzhong, 25 g sugar
Add the flour and salt. Mix on low speed until a shaggy dough forms, and there is no dry flour left. Cover and let the dough rest for 10 minutes. This rest helps the flour hydrate and makes kneading easier. 440 g all-purpose flour, 7 g salt
After resting, mix on medium speed for 5 minutes. Add the softened butter and continue mixing for another 8-10 minutes, until the dough is smooth, soft and elastic. 30 g unsalted butter
Transfer the dough to a lightly oiled container and cover.
Bulk ferment at room temperature until the dough has risen about 30 to 40%. At around 21 to 23 °C (70 to 73°F), this usually takes 3 to 4 hours, but timing will vary. The dough should look lightly puffed with a smoother, slightly domed surface and a few visible bubbles.
Once bulk fermentation is complete, cover the dough and place it in the fridge overnight.
To shape, divide the cold dough onto a floured bench into 12-14 equal pieces, about 65-75 g each. Shape each piece into a smooth ball, then roll the ball gently under your palm in one direction to form a short cone shape, slightly tapered at one end. This step helps create an even triangle later.
Flatten the cone gently, then roll it out into a triangle about 20-25 cm long (8 to 10 inches), with the wide end closest to you and the tip pointing away. The triangle does not need to be perfect. Dust the top of the dough with flour to stop it sticking to the rolling pin, but a little bit of friction between the dough and bench will help it roll out.
Cut the butter into short batons about 5 to 6 cm long (2 inches), roughly finger-shaped. Place one piece of cold butter across the wide end of the triangle. 120 to 145 g unsalted butter
Stretch the tip of the triangle gently to lengthen it slightly, then dust away any excess flour and roll the dough up tightly from the wide end towards the tip, enclosing the butter fully. Press the tip down firmly underneath so it does not unravel.
Place the shaped rolls seam side down on a lined baking tray, leaving space between them.
Cover loosely and proof at room temperature until slightly puffy and lighter to the touch, about 2-4 hours depending on temperature. They should still hold their shape.
Preheat the oven to 220C (430F) regular oven.
Just before baking, brush the tops with water and sprinkle lightly with flaky sea salt.
Bake until deeply golden brown, around 15 to 20 minutes depending on your oven. Butter will melt out during baking and bubble around the base, which is normal and gives the rolls their crisp bottom.
Remove from the oven and let the rolls sit on the tray for 5 to 10 minutes before serving. Brush the hot rolls lightly with melted butter after baking for extra shine and flavour. These are best eaten warm on the day they are baked.