Activate the yeast by combining warm milk (105-115°F) with 1 teaspoon sugar, sprinkle yeast over top, let stand 5-10 minutes until foamy.
Prepare mastic by freezing crystals for 10-15 minutes, then grind with a pinch of sugar until fine powder forms.
In stand mixer bowl, combine flour, remaining sugar, salt, ground mahlab, mastic, and orange zest. Whisk briefly to distribute evenly.
Add room-temperature eggs, activated yeast mixture, and vanilla. Mix on low speed for 3-4 minutes until rough, shaggy dough forms.
With mixer running on low, add softened butter one tablespoon at a time. Once incorporated, increase speed to medium and knead 10-12 minutes until smooth, glossy, and elastic. Perform windowpane test to confirm gluten development.
Place dough in oiled bowl, cover, and let rise in warm spot for 1.5-2 hours until doubled in size.
Turn dough onto lightly floured surface, gently deflate, and divide into 3 equal pieces (about 255g each). Roll each into an 18-20 inch rope.
Pinch ends of ropes together and braid three-strand style from top to bottom. Tuck ends underneath, transfer to parchment-lined baking sheet.
Cover loosely and let rise for 45-60 minutes until puffy but not doubled.
Preheat oven to 350°F (175°C). Whisk together egg and milk for egg wash. Brush entire surface of risen loaf gently.
Press red eggs into braids if using, sprinkle with sesame seeds or almonds. Bake for 30-35 minutes until deep golden brown. Tent with foil if browning too quickly.
Let rest on baking sheet for 10 minutes, then transfer to wire rack to cool completely (at least 30 minutes) before slicing.