Preheat the oven to 200 degrees F (95 degrees C). Grease a baking dish with oil.
Whisk eggs and sour cream together in a bowl. Season with salt. Mix in queso blanco, oil, and sparkling water. Stir in feta cheese. Mix in Parmesan cheese and ham until well combined.
Place 1 sheet of phyllo dough into the prepared baking dish and brush lightly with oil. Add the next phyllo dough sheet and brush with oil. Continue until you have used about 1/3 the phyllo sheets. Spread 1/2 the filling on top of the phyllo dough and smooth using the back of a spoon. Add the next 1/3 of phyllo dough sheets, brushing each sheet with oil. Spread remaining filling on top. Finish with the remaining 1/3 of phyllo dough. tucking edges into the sides of the baking dish.
Beat together 1 egg, 1 teaspoon salt, 1 teaspoon oil, and sour cream and gently pour mixture over the top of the pastry in the baking dish. Spread evenly.
Bake in the preheated oven until golden brown, about 35 minutes. Remove from heat and let sit 5 to 10 minutes before cutting.
