This is a delicious Greek bread that is traditionally made at Greek Easter and for New Year's. For Easter, the bread is baked with hard-boiled red eggs that accompany the feast celebrating the resurrection of Christ. —Maro
Proof your yeast. Add yeast and 1 tablespoon of the sugar to a 1-cup measuring cup with 1/4 cup of water. Set aside allowing yeast to foam and activate yeast for about 10 minutes.
In a small saucepan on low, melt butter and add milk. Transfer to a mixing bowl fitted with a dough hook and add the yeast mixture. Stir to dissolve the yeast. Add 1 1/4 pounds of the flour, then add the remaining sugar, 3 eggs, the mahlepi, and the masticha and mix until elastic enough to form a window when stretched. Add the remaining 1/4 pound of flour as needed, but keep in mind that it's a wet dough so it won't form a ball in the way other yeast doughs might. Mixing will take about 15 minutes total.
Transfer the dough to a large bowl. Cover dough and place in a draft-free environment and allow to rise for about 2 hours, until at least doubled in size.
Butter two 10-inch or three 9-inch round cake pans. Place dough on a floured surface to shape and divide according to the number of pans. Allow dough to rise again for about an hour, until again doubled in size.
Preheat oven to 325° F. Combine the remaining egg and the vanilla extract and brush tops of loaves with egg wash. Sprinkle with sesame seeds. Bake for about 25 to 30 minutes, until golden. Test with toothpick for doneness.