24 hours before, place the freezer bowl of your ice cream maker in the freezer to chill.
In a medium pot over medium-low heat, combine coconut cream, coconut milk, sugar, honey, cacao butter, vodka, vanilla bean paste, and salt. Stir as the solids begins to melt. When fully melted, remove the pot from heat and transfer to a bowl. Cover, and let cool completely at room temperature, or, for faster results, in an ice bath.
Transfer the mixture to a blender and add the basil leaves. Blend until completely smooth. Return the mixture to the bowl, and set in an ice bath to cool completely.
Place chocolate and coconut oil in a small saucepan over low heat and melt, stirring constantly. Once melted, remove from heat and set aside.
Freeze custard according to manufacturer’s instructions. After it's churned to the point of soft serve (about 15 minutes), pour in 3/4 of the melted chocolate in a thin stream. Chocolate will immediately harden into small bits and streaks. If churning is disrupted by pouring or clumping chocolate, simply transfer some of the churned ice cream into a freezer-proof container and set in freezer while you finish adding chocolate.
Transfer the ice cream to a freezer-proof container and freeze at least four hours, until set.
Serve with additional drizzles of melted chocolate on top.