Combine together the mince, salt and pepper and shape into 10 burger patties and chill for 30 minutes.
Heat the oil in a frying or griddle pan over medium high heat. Cook the burgers for 2 minutes on one side, then flip and top each one with two slices of cheese. Cook for a further 2 minutes until medium or 4 to 5 minutes if you want them well done. Remove from pan and rest for 2 minutes.
Halve the burger buns and brush with the melted butter. Grill until golden.
Spread the ketchup over the bun bases top with the cucumber, burger, American mustard and the lettuce, Spread the mayonnaise on the underside of the top bun and place on top. Serve with chips or wedges.