Step 1: Prepare the Phyllo Dough: Take the phyllo dough out of its package and tear it roughly into 1-2 inch pieces. Spread the pieces out on a large baking sheet or clean kitchen towels and let them air dry for at least 30 minutes, or up to 2-3 hours, until crisp and somewhat brittle.
Step 2: Make the Orange Syrup: In a medium saucepan, combine 2 cups of sugar, 2 cups of water, 1/2 cup of orange juice, and the zest of 1 orange. Bring to a boil over medium heat, stirring until the sugar dissolves. Reduce heat to low and simmer for 5-7 minutes until the syrup slightly thickens. Remove from heat and stir in 1 tsp vanilla extract. Let the syrup cool completely.
Step 3: Prepare the Cake Batter: Preheat your oven to 350°F (175°C). Grease a 9x13 inch baking dish. In a large bowl, whisk together the 4 eggs and 1 cup of sugar until light and fluffy. Add the Greek yogurt, olive oil, milk, zest of 2 oranges, and 1 tsp vanilla extract, mixing until just combined.
Step 4: Combine Dry Ingredients: In a separate bowl, whisk together the all-purpose flour and baking powder. Gradually add the dry ingredients to the wet ingredients, mixing until just incorporated. Do not overmix.
Step 5: Fold in Phyllo: Gently fold the dried, torn phyllo pieces into the cake batter until all pieces are coated. The batter will be thick.
Step 6: Bake the Cake: Pour the phyllo-batter mixture into your prepared baking dish, spreading it evenly. Bake for 45-55 minutes, or until the cake is golden brown on top and a wooden skewer inserted into the center comes out clean.
Step 7: Syrupy Perfection: Immediately after removing the hot cake from the oven, slowly and evenly pour the cold orange syrup over the entire surface. Let the cake rest at room temperature for at least 2-3 hours, or preferably overnight, to fully absorb the syrup. Slice and serve.