Add the flour, stiff levain, eggs, whole milk, sugar, and salt to the bowl of your stand mixer fitted with a dough hook. Mix until just combined. It does not matter if the dough is a bit lumpy at this point. Cover, and rest/autolyse for 30 minutes.
After the autolyse, continue mixing for 5 minutes.
After 5 minutes, you can slowly start to add cubed, softened butter. When all the butter is added, scrape down the sides of your bowl and continue mixing at medium speed for 15-25 minutes. Do the window-pane test to check if your dough is ready.
Transfer the dough to a clean bowl, cover, and proof in a warm spot (75°F-78°F/24°C-26°C) for 6 hours. If you don't own a dough proofer, you can keep the dough in the oven with the light turned on and the oven door slightly open.
I like to do two gentle folds during proofing, one after 1,5 hours and one after 3 hours.
After 6 hours, place the dough in the fridge and cold-proof for 6-8 hours.