For the sorbet:
To make the simple syrup, in a heavy medium saucepan, combine the sugar and water and bring to a boil over high heat. Reduce heat to medium-low, stir, and simmer until the sugar dissolves, about 5 minutes. Remove from the heat and cool slightly.
In a 2-quart saucepan, combine the coconut milk, water, lemongrass and ginger. Bring to a boil. Reduce heat to medium-low and simmer for 10 minutes, stirring occasionally. Remove from the heat. Add the simple syrup and stir well. Strain through a fine mesh strainer into a clean -container. Cool in the refrigerator for at least 2 hours. Pour into an ice cream machine and process according to the manufacturer’s instructions. Transfer to an airtight container and freeze until ready to serve.
In 4 tall, footed glasses, place 1 small scoop of coconut sorbet and pack the ice cream down into the bottom of the glasses with the back of a spoon.
Top with about 1 tablespoon of the chopped pineapple, about 1 1/2 teaspoons of the toasted coconut, and drizzle with about 1 tablespoon of the rum.
Continue with the remaining ingredients, so there are 4 layers of each, ending with pineapple and rum on top, pressing with a spoon to “seal” each layer.
Place the parfaits in the freezer for at least 30 minutes.
Remove from the freezer and top each with 2 tablespoons of whipped cream and garnish with white chocolate shavings. Serve immediately.
Yield: 4 servings