Thaw 160 g of your sourdough starter (if you keep it frozen) or take 160 g a few hours after you fed your starter.
Add the eggs, milk, sugar, salt, dry yeast, orange blossom water, orange syrup, and your starter to your stand mixer bowl and give it a good whisk.
Sift the flour and add it to the wet ingredients. Combine until there are no dry flour particles.
Let the dough rest for 20-30 minutes to allow the flour particle to hydrate.
Start kneading the dough at low speed for about 10 minutes or until the dough gets a bit elastic
Add the butter in 3 or 4 batches, making sure it's completely incorporated before you add the new batch.
Knead the dough at low speed until it's very elastic, shiny, the surface has blisters and it's not sticky. You shouldn't need to add more flour. It should pass the windowpane test
Cover the bowl and let the dough rise at room temperature until it doubles in size
Place the dough in the fridge for a cold fermentation for at least 15 h. You can leave the dough in the fridge for up to 2 days
Remove the dough from the fridge and gently deflate it. Cover it and let it rest for 10 minutes
Take 8 pieces of 130 - 140 g (4 - 5 oz) from the dough in equal pieces, shape them into balls and place then into your round baking pan.
Divide the rest of the dough into 2 equal pieces, flatten them with your hands and with a rolling pin roll them into a rectangle twice as wide as your rectangular baking pan.
Make a log with both pieces of dough, with a knife or a bench scraper divide them into equal halves, and place them in your rectangular baking pan.
Cover the two pans and let the dough rise to the rim of the pans.
Preheat your oven at 350 F
Before baking, whisk the egg yolk and the milk together and brush the brioche with it.
Bake the brioche for 45 minutes or until it's golden brown
For best results, let the brioche cool down to room temperature before cutting through it.