Trim sinew and most of the fat from the lamb and mince in a food processor. Finely dice shallot and chop mint; add these to the mince. Add the egg, salt, chilli and Armagnac.
Divide mix into 4 and roll into balls, and then slightly squash down. Cook the burgers in a frying pan with a little oil (or on the barbecue) for 4 minutes on each side. Slice cheese into four and put a slice of cheese on each burger.
Serve burgers on toasted Turkish bread with onion confit.