Grind the spices: Freeze mastic crystals for 10-15 minutes, then grind with a pinch of sugar into a fine powder. Grind mahleb seeds separately until very fine. Measure 1/2 teaspoon mastic and 1 tablespoon mahleb.
Bloom the yeast: Combine warm water with 1 teaspoon of sugar and active dry yeast. Let stand for 8-10 minutes until foamy and bubbly.
Warm butter-milk mixture: In a saucepan over low heat, combine milk, butter, orange zest, vanilla, mahleb, and mastic. Heat until butter melts and mixture is warm (100°F/38°C). Do not boil.
Mix the dough: In stand mixer bowl, combine remaining sugar, salt, 2 eggs, cooled butter-milk mixture, and yeast mixture. Add flour and mix with paddle until shaggy dough forms. Switch to dough hook.
Knead the dough: Knead on medium-high for 15-20 minutes until smooth, elastic, and slightly tacky. Do not add extra flour. The dough should be soft and pull away from bowl walls.
First rise: Transfer dough to a greased bowl, cover, and let rise in a warm place until doubled in size, about 2-4 hours.
Divide and rest: Gently deflate dough and divide into 6 equal pieces for 2 loaves or 3 for 1 large loaf. Cover and let rest for 10 minutes.
Shape and braid: Roll each piece into a 14-16 inch rope. Braid 3 strands together, pinching ends firmly. Tuck ends underneath. Repeat for second loaf. Press red-dyed egg into top if desired.
Second rise: Cover loaves and let rise for 45 minutes to 1 hour until puffed. Gently press side; it should spring back slowly.
Apply egg wash and bake: Preheat oven to 350°F (175°C). Brush loaves with egg wash (1 egg whisked with 1 tablespoon milk). Sprinkle with almonds or sesame seeds if desired. Bake 30-35 minutes until deep golden brown. Tent with foil if browning too quickly. Internal temperature should reach 190-195°F (88-90°C).
Cool completely: Transfer to wire rack and cool for at least 45 minutes to 1 hour before slicing to prevent gummy texture.