Place the minced beef in a food processor and add the chilli, garlic, onion, double cream and parsley. Season with a half a heaped teaspoon of salt and plenty of pepper, then blitz until the meat starts to hold together.
Divide the mixture into four and shape around the cubes of Cheddar into 10cm flat discs, either by hand or pressing into a metal pastry cutter. Arrange on a flat plate, then cover loosely with clingfilm and chill for at least 1 hour to allow the mixture to firm up.
Preheat the grill or a barbecue with medium-hot coals and cook the burgers for about 5 minutes on each side for medium.
Meanwhile, place the mayonnaise in a bowl and mix in the lettuce, gherkins, onion and mustard. Season to taste and mix well to combine.
Lightly toast the cut sides of the rolls under the grill or on the barbecue for 1-2 minutes. Cover the bottom half with a couple of slices of tomato and then sit the burger on top. Add a large spoonful of the lettuce mayonnaise to each one and cover with the top half of the bun. Serve immediately.