Step 1
Pour yeast into a small bowl; add warm water and 1/4 cup of flour. Stir cover with two damp kitchen towels and leave in a warm place until bubbly about 2 1/2 hours.
Step 2
Combine butter shortening and mastic in a saucepan over medium-low heat.
Step 3
Stir milk and sugar together in another saucepan over medium-low heat until sugar is dissolved and mixture is warm. Remove from heat.
Step 4
Pour 2 cups flour into the butter mixture and stir well. Add salt then stir in the yeast mixture. Add 2 more cups of flour and stir well. Stir in 3 beaten eggs. Slowly add the milk mixture and 1 cup of flour and stir well. Add the room-temperature water and the last cup of flour and stir until the dough is not sticky.
Step 5
Knead dough on a lightly floured surface for 20 minutes with lightly floured hands. Form dough into a ball. Cover with damp towels and let rise in a warm place until doubled in volume about 2 hours.
Step 6
Preheat the oven to 325 degrees F (165 degrees C). Grease two baking sheets.
Step 7
Divide dough into two pieces and separate each piece into three strands. Braid each of the 3 strands together to form 2 braided loaves. Place on the baking sheets. Nestle the red-dyed eggs between the strands. Brush the dough with the remaining beaten egg. Sprinkle the tops with almonds and sesame seeds.
Step 8
Bake in the preheated oven until golden brown about 25 minutes.