Finely grate the zest from 4 oranges and set aside. Squeeze the oranges to get about 1.5 cups of juice. Cut 1 additional orange into slices.
In a saucepan, combine the water, freshly squeezed orange juice, sugar for the syrup, orange slices, and cinnamon sticks. Bring to a boil and cook for a total of 10 minutes. As the syrup boils, skim off any foam that rises to the surface using a fine-mesh strainer.
Remove the syrup from the heat and set aside to cool completely.
Preheat the oven to 300°F (150°C). Slice the phyllo dough into strips. Gently separate the strips with your hands and toss them loosely and fluffily into a baking pan, making sure they are not packed together.
Bake the phyllo dough for 10 minutes, then gently turn and toss it so that it dries evenly. Return it to the oven and bake for another 10 minutes.
Remove the phyllo from the oven and set aside to cool. Once cooled, gently crush the dried phyllo with your hands into smaller pieces, being careful not to crumble it too finely.
Preheat the oven to 340°F (170°C). In a large bowl, combine the cake sugar and extra virgin olive oil and mix well.
Add the eggs and whisk until combined. Add the Greek yogurt, baking powder, and the reserved orange zest, and mix again until smooth.
Add the dried, crushed phyllo to the mixture and gently fold until all the pieces are evenly coated with the batter.
Grease an approximately 9 × 12-inch (22 × 30 cm) baking pan. Pour the mixture into the prepared pan and spread it evenly. Bake for approximately 30 minutes, or until the portokalopita is beautifully golden.
As soon as you remove the hot portokalopita from the oven, pour the cooled orange syrup evenly over the entire surface. Set aside for approximately 15 minutes to absorb the syrup before serving.
Cook it hands-free.
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