Step 1
Prepare a barbecue for high heat. Coat the capsicum with oil. Barbecue capsicum, turning occasionally, for 15 mins or until skin is charred all over and capsicum is tender. Transfer capsicum to a medium bowl, cover with plastic wrap and set aside until cool enough to handle. Peel and seed the capsicum, then finely chop.
Step 2
In a medium bowl, mix the cheddar cheese, cream cheese, 1 tbs of the mayonnaise, cayenne pepper and 1/4 cup of the chopped capsicum. Season with salt and pepper.
Step 3
Divide the mince into 4 portions. Shape into patties that are just slightly larger in diameter than the buns. Season with salt and pepper. Using your thumb, make a small imprint in the centre of each patty. Barbecue the patties for 3 mins. Turn patties and barbecue for 2 mins. Spread cheese mixture over the patties. Barbecue, covering barbecue to melt the cheese, for 1 min or until cooked through. Transfer patties to a plate. Set aside for 5 mins to rest.
Step 4
Meanwhile, in a medium bowl, toss the cabbage and onion with the remaining 1/4 cup (75g) mayonnaise and buttermilk. Season with salt and pepper.
Step 5
Top bun bases with slaw, patties and bun tops. Serve with lime halves.
Make it ahead: Make the cheese mixture up to 3 days ahead, cover and store in the fridge. Use leftover capsicum in salads. Make the patties up to 1 day ahead, cover and store in the fridge.