1. Process zest, sugar, and salt in food processor until damp, ten to fifteen 1-second pulses. With machine running, add lime juice/water mixture in slow, steady stream; continue to process until sugar is fully dissolved, about 1 minute. Strain mixture through nonreactive fine-mesh strainer into medium bowl; stir in Triple Sec, then cover with plastic wrap and chill in freezer until very cold, about 40 degrees, 30 to 60 minutes. (Alternatively, set bowl over larger bowl containing ice water.) Do not let mixture freeze.
2. When mixture is cold, using whisk, whip cream in medium bowl until soft peaks form. Whisking constantly, add juice mixture in steady stream, pouring against edge of bowl. Immediately start ice cream machine and add juice/cream mixture to canister; churn until sherbet has texture of soft-serve ice cream, 25 to 30 minutes.
3. Remove canister from machine and transfer sherbet to storage container; press plastic wrap directly against surface of sherbet and freeze until firm, at least 3 hours. (Can be wrapped well in plastic wrap and frozen for up to one week.) To serve, let sherbet stand at room temperature until slightly softened and instant-read thermometer inserted into sherbet registers 12 to 15 degrees.
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