Trim and discard root ends and tough tops from lemon grass; remove and discard tough outer layers. Coarsely chop tender inner stalks.
In a 2- to 3-quart pan over high heat, bring 2/3 cup water to a boil. Stir in lemon grass, remove from heat, and let stand for 15 minutes. Press through a fine strainer set over a bowl, then return liquid to pan; discard lemon grass.
Add sugar to lemon grass water and stir often over high heat just until dissolved, 2 to 3 minutes. Remove from heat and stir in lemon peel and juice. Nest pan in a bowl of ice water and stir often until mixture is cold, about 10 minutes. Stir in buttermilk and half-and-half.
Pour mixture into an ice cream maker (1-qt. or larger capacity) and freeze according to manufacturer's directions until dasher is hard to turn or machine stops. Package airtight and freeze until firm enough to scoop, at least 2 hours or up to 2 weeks.