Preheat the oven to 350 degrees F (175 degrees C). Lightly grease a 9x13-inch baking pan with oil.

Remove phyllo sheets from the package. Use a sharp knife to cut through the sheets, forming 1/4-inch strips. Lightly separate the strips with your fingers and place on a sheet pan in an even layer. Place in the oven while it preheats until phyllo has dried out and is beginning to turn golden, about 10 minutes. Remove and let cool.

Meanwhile, make the syrup: Combine sugar, water, and zest of 2 lemons in a saucepan over medium-high heat. Bring to a boil and simmer for 8 minutes. Remove from heat and stir in lemon juice from all 4 lemons. Allow syrup to cool while you prepare the cake.

Combine yogurt, oil, sugar, eggs, vanilla extract, baking powder, and the remaining lemon zest in a large bowl. Whisk until evenly combined. Add the dried and cooled phyllo to the bowl, crushing lightly. Stir everything together until yogurt mixture is evenly distributed, ensuring there are no dry spots of phyllo. Add mixture to the prepared baking pan and spread to an even layer.

Bake in the preheated oven until the top is golden and the filling set, about 45 minutes.

Remove lemonopita from the oven and immediately ladle the cooled syrup over the hot cake. Set aside until most of the syrup has fully soaked in, at least 4 hours. Slice into squares and serve.