PREPARE YOUR ICE CREAM MAKER
The night before you plan to make the ice cream, freeze the bowl of your ice cream maker.
MAKE THE ICE CREAM
Shake the cans of coconut milk before opening. In a blender, combine the basil with the coconut milk, honey, cornstarch, and salt. Blend until completely smooth, and transfer to a large stockpot. Turn the heat to medium and add 1 teaspoon of the lime zest and 1 tablespoon of the lime juice. Bruise and bend the lemongrass stalk, slightly separating the tubes from each other, and add to the pot. Stir to distribute. Heat over medium heat, stirring, until the mixture thickens enough to coat the back of a spatula. Do not boil.
Remove from the heat, cover, and let the mixture steep 30 minutes to 1 hour, longer for a stronger flavor. Once it has steeped, strain out the lemongrass and stir in the remaining ½ teaspoon lime zest and 1 tablespoon lime juice, and add the vodka, if desired.
Process in the ice cream maker, according to the manufacturer’s instructions, until frozen. Pour the frozen ice cream into a freezer-safe pan or dish and freeze for at least 6 hours before eating.
MAKE THE BRITTLE
Preheat the oven (or toaster oven) to 425°F. Place a piece of parchment paper on a small rimmed baking sheet. Combine all the ingredients in a small bowl and mix until the sesame seeds are thoroughly coated. Spread in a thin, even layer on the parchment paper and bake for 5 minutes, checking at 3 minutes to make sure nothing burns. When the brittle is bubbly and fragrant, remove it from the oven and let cool completely. Break into pieces and store in an airtight container.
Serve the ice cream topped with the black sesame brittle and a flourish of extra lime zest.
Note: If you don’t have an ice cream maker, pour the prepared mixture into a freezer-safe pan or dish, cover, and place in the freezer. After 2 hours, or when the mixture begins to freeze, remove it from the freezer and stir vigorously to break up the frozen sections. Repeat every 30 minutes until frozen, another 2 to 3 hours. You can do this using a whisk or spatula, an immersion blender, or an electric mixer.