Place the mince in a bowl with the ketchup, mustard and chives. Mix really well together, working the mixture with a fork to grind and blend the ingredients. Shape into 8 patties and place on a plate. Cover with clingfilm and chill for 30 minutes to firm.
Heat the oil in a large frying or griddle pan and cook the mushrooms over a high heat for 3–4 minutes until golden and soft. Remove from the pan using a slotted spoon. Add the burgers and cook over a moderate heat for 4–5 minutes on each side until golden and cooked through. Sit a Gruyère slice on top of each of the burgers and cover with a baking sheet for 1 minute to allow the cheese to soften.
Top the base of each bun with a burger, then spoon over the mushrooms and spread the top half of each bun with ketchup or a sauce of the child's choice. Place the lids on, and serve.