Flatten the chicken, by bashing it with a rolling pin. Dip the chicken into the beaten egg and then coat with the breadcrumbs.
Sallow fry on both sides in a frying pan over a medium heat for 3 to 4 minutes until both sides are golden brown and the chicken is no longer pink in the centre. Transfer the cooked chicken to a small ovenproof baking dish.
Make the bechamel sauce by melting the butter in a saucepan, then add the flour and stir until mixed. Gradually add the milk and whisk by hand until a nice smooth consistency forms, add more milk if needed.
Pour the thick sauce over the chicken and sprinkle with the grated cheese.
Place the chicken under a hot grill until bubbling hot and the cheese has melted. Serve with a nice green salad, and fries.