Heat oil in a skillet over medium heat; cook and stir onion until golden brown, 10 to 15 minutes. Add vegetables, lemon juice, and 1 pinch salt to onion; stir well. Remove skillet from heat.
Mix eggs, feta cheese, milk, mozzarella cheese, salt, and pepper together in a separate bowl.
Mix cold water and corn oil together in a bowl. Place flour and 1 pinch salt in a separate bowl. Slowly pour oil mixture into flour mixture, mixing until a thick dough forms. Press the dough into two 9-inch round baking pans. Spoon the vegetable mixture into the crusts and top with egg mixture.
Bake in the preheated oven until quiches are set in the middle, about 20 minutes.