Heat oil in deep-fryer to 190 degrees C. Preheat oven to 120 C / Gas mark 1/2.
Blend eggs and milk together in a large bowl. Place breadcrumbs in another large bowl. Dip aubergine slices first into egg mixture, then breadcrumbs, coating well. Fry in batches in hot oil, turning to brown evenly, then transfer to a kitchen roll-lined dish to drain. Repeat process for the chicken, lightly browning but not overcooking.
Generously grease a large, deep baking dish. Working in single layers, arrange aubergine, then chicken, then grated mozzarella cheese, until all ingredients have been used. Pour cheese sauce evenly over the top.
Bake in the preheated oven for 30 minutes. Increase heat to 180 C / Gas 4, and continue baking for another 20 minutes. Serve hot.