To make the syrup, cut a 2-inch section from the thickest part of the lemongrass stalk (near the base), trim the end and remove the thick, outer layer. Combine with the ginger and 1 1/2 cups water in a small saucepan and bring to a boil. Lower the heat, simmering for 5 to 10 minutes until reduced by about half, and strain through a fine sieve. Return the liquid to the saucepan, add 1/3 cup sugar and bring to a boil, stirring just until the sugar dissolves. Lower the heat and simmer gently until the liquid becomes syrupy, about 5 minutes. Let cool. The syrup may be stored in the refrigerator in an airtight container for up to three days.
To make the pudding, bring the milk and coconut milk to a simmer in a medium, heavy saucepan over medium heat. Stir in the rice and remaining 1/2 cup sugar. Return to a boil, lower the heat and simmer uncovered, stirring occasionally, for 20 to 25 minutes, until thick and creamy. (The rice should be a touch al dente.) Serve warm or chill, covering the surface of the pudding with plastic wrap to prevent a skin from forming. To serve, spoon the pudding into small bowls and drizzle some of the syrup over the top. If you like, garnish with slices of mango or papaya and fresh mint leaves.