In a bowl, mix together the patty ingredients with your hands. Gently shape the mixture into four patties of equal size and about 2.5 cm thick. With your thumb or the back of a spoon, make a shallow indentation about 2.5 cm wide in the centre of each patty to prevent it from doming as it cooks. Refrigerate the patties until ready to barbecue.
To make the mayo, whisk together all the ingredients in a small bowl. Cover and refrigerate until needed.
Prepare the barbecue for direct cooking over medium heat, 180–230°C (350–450°F).
Brush the cooking grills clean. Brush the patties on both sides with oil. Barbecue the patties over direct medium heat for 11–13 minutes, with the lid closed, until fully cooked, 74°C (165°F), turning once. During the last 30 seconds–1 minute of cooking time, place a slice of cheese on each patty to melt, and toast the rolls, cut-side down, over direct heat.
Build a burger on each roll with some mayo, a lettuce leaf and a patty. Serve warm.
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