Roughly chop the chorizo and put the pieces in a food processor with the chopped half of red onion and pulse a few times. Add the beef mince and a big pinch of seasoning, then pulse again until just combined. Be careful not to over-mix or you’ll be left with a soggy paste. Shape the mince mixture into 4 even, chunky patties. (The chorizo burgers will keep well, covered and chilled, for up to 24 hours. Bring them up to room temperature before cooking.)
Brush the burgers with the oil (this will prevent them from sticking) and barbecue or griddle for about 8 minutes, turning regularly, until browned and cooked through, but still juicy.
To make the aioli, combine the mayonnaise, garlic and a pinch of salt in a small bowl. Give everything a good mix, then set aside until needed.
Put the burger buns, cut side down, on the barbecue or under a grill until lightly toasted and golden. To serve, divide the aioli between the toasted burger buns and top each with the peppers, a chorizo burger, some red onion rings and rocket.
For top tips on protecting you and your family when preparing raw meat and poultry, visit Food Safety in the Home and view the full Food safety at home video.