Make simple syrup by combining granulated sugar and water in a saucepan, then heat over medium while stirring until the sugar is fully dissolved, about 3-4 minutes. The mixture should look clear with no visible sugar grains.

Remove the saucepan from heat and let the simple syrup cool completely, then stir it once to keep it uniform. Chill readiness looks like room-temperature syrup with no warmth.

Combine the cooled simple syrup with fresh lemon juice, lemon zest, and salt, then stir until evenly blended. The mixture should turn a pale, concentrated yellow and smell sharply of lemon.

Refrigerate the lemon mixture until very cold, at least 4 hours. It should feel icy-cold throughout with no neutralized warmth remaining.

If using egg white, whip it to soft peaks, then fold it into the cold lemon mixture gently. It should look slightly lighter and airy, with streaks disappearing.

Churn in an ice cream maker until thick and slushy, about 20-25 minutes. The sorbet should visibly thicken as it turns in the machine.

Transfer the churned sorbet to a container, cover, and freeze at least 2 hours until firm. It should hold shape when scooped and look fully set around the edges.

Serve in chilled bowls or scoop into hollowed lemon halves. The surface should look lightly frosted and the texture should be scoopable immediately.