Process the orange zest, granulated sugar, and salt in a food processor until damp, ten to fifteen 1-second pulses.
With the machine running, add the fresh-squeezed orange juice and lemon juice in a slow, steady stream; continue to process until the sugar is fully dissolved, about 1 minute.
Strain the mixture through a fine-mesh strainer into a medium bowl.
Stir in the Triple Sec, then cover with plastic wrap and chill in the freezer until the mixture reaches 40 degrees F, about 30 to 60 minutes.
When the mixture is cold, whip the heavy cream in a large bowl until soft peaks form.
Continue to whip constantly and add the juice mixture in a slow, steady stream, pouring against the edge of the bowl.
Immediately start the ice cream machine and add the orange mixture to the canister; churn according to the manufacturer directions.
Transfer the sherbet to a freezer-safe container and freeze until firm, about 3 hours.
