Remove pineapple leaves; reserve for garnish. Peel, core and cut pineapple into chunks.
Mix sugars, pineapple juice, cinnamon sticks and nutmeg in heavy large saucepan or Dutch oven. Cook on medium heat until mixture comes to boil. Add pineapple chunks; cook and stir 35 minutes or until pineapple is golden brown. Let cool to room temperature. Remove cinnamon sticks.
Place pineapple mixture, ice, mezcal, lime juice and vanilla in blender container; cover. Blend until smooth. Pour into beverage glasses. Garnish with pineapple leaves.