Add 500g of bread flour, 55g of granulated sugar, 6g of instant dry yeast, 1 tbsp of honey, three whole eggs, and 175g of warm milk, 100g room temperature water in a bowl of a stand mixer equipped with a dough hook attachment.
Turn on the mixer at low speed to avoid spilling the flour and mix until well combined.
After that, we can add the rest of the ingredients.
Grate 1 orange zest and lemon zest. Add it to the bowl and 1 tbsp of vanilla extract and 10g of kosher salt.
Mix the dough at medium speed until it forms a tight ball and separates from the bowl edges.
At this point, we can add olive oil. In the past, in Italy, it was almost impossible to find butter, but they were plenty of olives; that's where the use of olive oil comes from. That's the main difference between the french and Italian brioche.
Add the olive little by little, only adding more after completely incorporating the previous one.
This step is fundamental to keep the strength of the dough and avoid collapsing while baking it.
Keep mixing until the bottom and sides of the mixer are completely clean, and the dough is a nice and smooth ball.
Turn off the mixer, sprinkle some flour on the work surface, and place the dough on it.
The dough is still slightly sticky, so we'll fold it a few times until it is perfectly smooth.
Move the dough into a lightly oiled bowl, cover with cling film and let rest the dough at room temperature for 1 hour to start the fermentation process, then place in the fridge overnight.
The next day, grab the dough from the fridge, which should be doubled its size, and deflate it by pressing with your hands.
Place the dough on a lightly floured work surface and divide the dough into 20 equal pieces of 50g each.
Working with one piece of dough at a time, flatten out the piece of dough, then tuck it up into a ball, then turn the ball seam side down and roll it into a tight ball by cupping your hand to create a 'claw' shape, using the tension from the counter to roll the dough tightly.
Repeat the process until you roll all the pieces of dough and place them in a baking tray lined with parchment paper and cover them with a kitchen towel.
Let them proof in a warm place for a couple of hours or until jiggly and doubled in size.
Prepare an egg wash with one egg yolk mixed with one tablespoon of water and brush each bun with it, being gentle to avoid deflating the bun.
Bake the buns at 370°F for 8 to 10 minutes or until they are nice and golden on the top.
After baking and when still hot, brush them with simple syrup to give them a glossy look.
After that, transfer them to a wire rack and let them cool for 20 minutes.