Combine sugar, lime zest, and lime juice in a heatproof bowl. Add 4 whole basil leaves.

Using the back of a large spoon, bruise the basil leaves against the bottom or side of the bowl.

Add boiling water and stir until the sugar has dissolved. Cover and let steep for 15 minutes.

Discard the basil leaves. Cover and refrigerate until chilled, about 1 hour or overnight.

Slice remaining basil leaves into very thin strips. Add sliced basil and yogurt to the lime mixture and whisk until smooth.

Pour into an ice cream maker and freeze according to manufacturer’s directions.

Alternatively, freeze the mixture in a shallow metal pan until solid, about 6 hours. Break into chunks and process in a food processor until smooth.

Serve immediately or transfer to a storage container and let harden in the freezer for 1 to 1½ hours. Serve in chilled dishes.