In a bowl, mix 2 of the eggs, olive oil, milk, water, and 1 teaspoon of the salt with a whisk. Set aside.

Saute onions in 1 tablespoon of the butter until tender. Remove and cool. Combine cooled onions, spinach, kale, mint, feta, kasseri and remaining eggs. Season with salt and pepper and mix well. Grease a 9- by 13-inch casserole dish with ½ tablespoon of the remaining butter. Melt the remaining butter and add it to the milk mixture and whisk to combine.

Place two sheets of the pastry in the bottom and over the sides of the dish. Spread 2-3 tablespoons of the liquid mix on top. Take another two sheets of pastry, fold in half, and place in the bottom of the dish over the phyllo. Continue layering phyllo in this manner, topping each layer of 2 folded sheets with 2 to 3 tablespoons of the liquid mixture, until halfway through the pastry. Then, place the entire spinach-cheese filling onto the phyllo stack. Resume layering the pastry and the liquid mix as before, over the filling, until you have used all of the pastry. Then, fold the sides of the bottom layer over the top. Pour the remaining liquid over the top and spread evenly to cover the phyllo. (This will prevent it from burning in the oven.)

Cover the casserole with plastic wrap and refrigerate for 2-3 hours.

Preheat oven to 350 degrees F. Bake the casserole, uncovered, until golden brown, 45 to 50 minutes. Remove and allow to cool slightly before serving. May be served warm or at room temperature.