Mix all ingredients up to and including powdered sugar in a bowl. Add the zest of the lime and 3 tablespoons of the juice, stir again and refrigerate for at least 1 hour. Then freeze in the frozen (stir vigorously 3 - 4 times) or prepare in the ice cream machine.
Peel the pineapple, cut out the stalk and then cut into slices or cubes.
Grill over medium, direct heat for 2 to 3 minutes on each side, best to place the cubes on 8 (watered) wooden skewers.
Pluck the herbs, sprinkle with the sugar and finely chop everything.
Serve the ice cream in glasses, turn the pineapple skewers in the herb sugar and pour onto the glasses.