In a frying pan, fry the onion gently in 1 tsp oil for about 5 mins or until really soft. Put into a bowl and leave to cool.
Add the mince, cheese, egg and some freshly ground black pepper, and mix together.
Shape the mixture into 2 large burgers for two adults and 2 smaller burgers for two toddlers. (You could freeze them now if you wanted to or just put them onto a plate, cover, and put into the fridge.)
Heat the rest of the oil in the frying pan and fry the burgers or grill under a hot grill for 4-5 mins each side for the large burgers and 3-4 mins each side for the smaller burgers, or until cooked through.