In a small bowl, add the yeast, 1/2 cup lukewarm water, and 1 teaspoon of sugar and stir together. Set it aside and let it rest for 5-10 minutes to activate the yeast and let it bloom.
Separate the egg yolks and egg whites.
In the bowl of your stand mixer using the paddle attachment, beat the butter and oil together (be sure to start on low speed to avoid splattering oil everywhere). Add the egg yolks and sugar and mix until light and fluffy.
In a separate bowl, beat the egg whites until they form stiff peaks. Add the egg whites to the butter mixture along with the yeast mixture, mahlepi, mastiha, warm milk, orange juice, orange zest and salt and mix until combined.
Replace the paddle attachment with the dough hook on your stand mixer. Slowly add the flour and mix the dough on low speed for 10 minutes until the dough is smooth and elastic and pulls away from the side of the bowl. The dough will be slightly sticky at this point.
Cover the bowl with plastic wrap and clean kitchen towels and place it in dry, warm spot. Let it rise until it doubles in size (about 2 hours).
Once the dough has risen, gently deflate it and then divide the dough into 2 portions to make 2 large tsoureki loaves or 4 portions if you are making smaller loaves.
Working with one portion at at a time, divide it into 3 equal dough balls. Roll each of those balls into a long cylindrical log shapes using your hands. Braid the 3 logs together to form the braided loaf, pinching the ends together lightly and tucking them underneath the loaf. Repeat with the other portion(s).
Transfer the loaves onto a baking sheet and let them rise again for 1 hour.
Preheat oven to 350F.
In a small bowl, whisk the egg and water to form the egg wash. Brush the egg wash over the loaves and sprinkle with almond slivers.
Bake for 25-30 minutes or until the loaves have browned nicely. Transfer to a cooling rack to cool completely.