In the bowl of a kitchen aid mixer, combine 500g flour, 300g water, the tangzhong, the ripe levain, salt, and maple syrup.
Fit the mixer with the paddle attachment and set to STIR for 1-2 minutes.
Scrape down the sides of the dough with a rubber spatula (the dough will be pretty wet), and switch to the dough hook attachment.
Using the dough hook, mix on low or medium low (I set it between 2 and 4 on my mixer) speed for 7 minutes. Pause once more in the first couple minutes to scrape down the sides one more time. The dough should start to come together and cling to the dough hook rather than sticking to the sides of the bowl
Turn off the mixer and let the dough rest for 10-20 minutes in the bowl
In the meantime, check that your butter is nice and soft and your finger can easily leave an indent in one of the pieces. If it's still a bit firm, you can put it in a microwave for a couple seconds.
Once the dough has rested in the bowl and the butter is ready, turn the mixer back to a low speed and slowly add the butter one piece at a time, allowing several seconds between pieces so that each piece has time to incorporate into the dough. This should take about 5-7 minutes.
When the butter is fully incorporated, turn up the speed slightly and continue mixing for 1-2 more minutes.
Transfer the dough to a separate bowl for bulk fermentation (or leave it in the mixer bowl).
Cover the bowl and allow the dough to bulk ferment for 3-4 hours until double in size.