In a small bowl, dissolve the yeast in the lukewarm milk with a teaspoon of sugar. Let it stand for about 10 minutes until it becomes frothy.
In a large mixing bowl or stand mixer fitted with a dough hook, combine the flour, sugar, salt, orange zest, mahleb, and mastic gum if using.
Make a well in the center and add the yeast mixture, melted butter, eggs, and vanilla extract.
Begin kneading on low speed or by hand, gradually incorporating the flour into the wet ingredients. Knead until the dough is soft and elastic, about 12-15 minutes. It should be slightly sticky but manageable.
Lightly oil a clean bowl, place the dough inside, and cover with a clean kitchen towel or plastic wrap. Let it rise in a warm place until doubled in size, about 2 hours.
Once risen, punch down the dough gently and divide it into three equal pieces. Roll each piece into a long rope about 40-45 cm in length.
Braid the three ropes together, tucking the ends underneath to secure them, forming a classic three-strand braid. Carefully transfer the braid onto a lined baking sheet.
Nestle the red-dyed egg into the center of the braid, pressing gently to make a small indentation.
Cover the loaf loosely with a towel and let it rise again for 45-60 minutes until puffy.
Preheat your oven to 175°C (350°F) while the bread is rising.
Beat the remaining egg with a splash of water and gently brush it over the entire surface of the bread, taking care not to brush the red egg.
Sprinkle the braided loaf generously with sliced almonds and sesame seeds.
Bake in the preheated oven for 30-35 minutes, or until the bread is a deep golden brown and sounds hollow when tapped on the bottom.
If the bread browns too quickly, loosely cover it with aluminum foil halfway through baking.
Remove from the oven and cool on a wire rack before serving.