Blend the sugar, mastic, and mahleb in a food processor until finely ground. Transfer the mixture to a large bowl, then add the buttermilk, olive oil, and yeast. Stir and set aside for 5 minutes.
Combine the flour, salt, and orange zest in the bowl of a stand mixer. Mix with a spoon, then add the yeast mixture and knead with the dough hook on medium to high speed for about 4 minutes.
Add the egg and continue kneading for an additional 6 minutes. Once the dough is ready, transfer to a large greased bowl, cover with plastic wrap, and leave to rise for 1.5 to 2 hours.
Once the dough has risen, transfer to a floured surface and roll out into a rectangle about 37 x 30 cm (14.5 x 12 in). Use a glass to cut out circles with a diameter of 7 cm (~3 in), and use the lid of a water bottle to cut out holes in the center of each circle.
Place the dough circles on a baking sheet lined with parchment paper, cover with plastic wrap, and let them rest for 30 to 60 minutes.
Heat a pan over medium to high heat and fill with sunflower oil to just above halfway. Fry the donuts in batches for 10-12 minutes, turning them occasionally, until golden brown on both sides and cooked through. Transfer each donut to a tray lined with paper towels to drain. Allow them to cool completely.