Preheat grill to medium-high heat. Grease the grill rack with vegetable oil.
Season chicken fillets with salt and pepper on all sides. Place on grill and cook 5-6 minutes per side, brushing with lime juice occasionally, until the internal temperature reaches 74°C (165°F).
Place pineapple slices on grill and cook 2-3 minutes per side until charred.
Toast the burger buns in a toaster or under a broiler. Spread mayonnaise (if using) on the inside surfaces of each bun. Assemble each sandwich by placing one chicken fillet, one pineapple slice, and one basil leaf between the two halves of the bun.