Preheat the oven to 170°C (340°F), fan mode (or 180°C / 350°F conventional).
In the bowl of a stand mixer, add the lukewarm milk, yeast, and sugar. Whisk to combine and let the mixture sit for 5 minutes, until the yeast is activated and slightly foamy.
Add the flour, salt, mahleb, mastic, orange zest, and eggs. Using the dough hook attachment, knead on medium to medium-high speed for about 5 minutes, until a smooth dough starts to form.
Increase the mixer speed slightly and gradually add the butter, a little at a time. Continue kneading for 5–6 more minutes, until the dough becomes smooth, elastic, and slightly sticky.
Lightly grease a mixing bowl with olive oil and transfer the dough into it. Cover tightly with plastic wrap and let it rise in a warm place for about 1 hour, or until doubled in size.
Once risen, divide the dough in half. Take one half and divide it into 4 equal pieces. Roll each piece into a rope about 15 cm (6 inches) long.
Pinch the ends of the ropes together and braid them into a neat loaf.
Transfer the braided loaf to a baking sheet lined with parchment paper. Repeat the same process with the remaining dough to make 2 loaves total.
Cover the loaves with a kitchen towel and let them rise again for 30–40 minutes, until slightly puffed.
Brush the tops with egg yolks and bake for 20–30 minutes, or until beautifully golden brown.
Remove from the oven, allow to cool for 5 minutes, and serve.