Make the labneh the day before
Place a cheesecloth or a clean, rinsed kitchen towel in a sieve resting on a bowl and pour the yogurt into it. Fold the cloth closed and let it drain overnight in the refrigerator.
The next day, you'll have about half the amount left: a thick, creamy labneh that is perfect as a filling.
Making brioche dough
Combine the lukewarm milk with the yeast and a small amount of the sugar in a measuring cup. Let stand for 5 minutes until it starts to foam.
Meanwhile, put the flour, the rest of the sugar, and the salt in the bowl of your stand mixer. Add the cold butter cubes, the eggs, the egg yolks, and the yeast mixture.
Knead the dough for at least 10 minutes. This long kneading creates a lot of gluten, resulting in that delicious airy brioche texture.
Turn the dough out onto a lightly floured work surface, form a ball, and wrap it in plastic film. Let it firm up in the refrigerator for a minimum of 4 hours.
Forming the buns
Divide the dough into 6 equal pieces.
Meanwhile, butter and flour a muffin tin.
First, form each piece of dough into a ball and then roll it out into an oval shape, about 12 cm long. Sprinkle with za'atar and fold the dough in half onto itself.
Using a sharp knife or dough scraper, make incisions every half centimeter on the folded side. Then roll up the dough and place it upright in the muffin tin.
Repeat with the other pieces of dough.
Let the buns rise until they are nicely airy and doubled in volume.
Baking
Mix the egg yolk with a small splash of milk and brush the buns with it.
Bake for 25 minutes in a preheated oven at 170°C until golden brown and nicely risen.
Let cool completely.
Making the labneh filling
Mix the drained labneh with pepper, salt, olive oil, and 1 tablespoon of za'atar.
Put into a piping bag.
Make a small hole at the bottom of each brioche bun and fill generously with the labneh.
Pipe a small dollop of extra labneh on top. Using a spoon, make a small dimple in the filling, drizzle with some olive oil, and finish with extra za'atar.
Optionally, garnish with thyme flowers for an extra beautiful finishing touch.