Mix together all the ingredients for the burgers. Divide the mixture into four portions. Form each portion into a patty about half an inch thick. Lay the patties on a plate, cover the plate with cling wrap and chill the burgers while you prepare the relish.
Thinly slice the onions and bell peppers.
Grate the garlic.
Heat the olive oil in a pan. Add the sliced onions and bell peppers, and basil. Cook, stirring oven, just until the onions and peppers start to turn soft.
Pour in the vinegar. Add the sugar, salt, pepper and garlic. Stir. Cover, turn down the heat to lowest setting and slow cook until the vegetables are very soft and the mixture is almost dry. The vegetables will expel water and that, in addition to the rice wine vinegar, should be just enough liquid.
Cover, turn down the heat to the lowest setting and slow cook until the vegetables are very soft and the mixture is almost dry, about 15 to 20 minutes.
If there’s still too much liquid by the time the vegetables are soft, turn up the heat to medium and cook, uncovered, for a few minutes, stirring often.
Cool the relish. And while it cools, grill the burgers then split and toast the buns.
To serve: Spread the bottom halves of the buns with mayonnaise. Drop in the burgers. Top with about two tablespoons of the onion and pepper relish and serve.