To prepare Jerk Marinade, combine all ingredients into a food processor and puree until smooth.
To prepare the hamburger patties, combine meat with ¼ cup of marinade, the egg and breadcrumbs. Form into 4 patties, cover and chill in fridge for at 30 minutes for patties to firm up.
Place the pineapple rings in a shallow dish and spoon about ¼ cup of the marinade over, turning to coat. Let sit for 30 minutes.
To prepare the aioli, combine all ingredients stirring until smooth.
Brush patties and pineapple rings with a bit of oil and grill until meat is fully cooked, and rings are warm and marked on both sides.
To serve, toast the hamburger buns if desired, spread on some aioli and top with a pineapple ring, some lettuce and a pattie.