In a medium saucepan, combine the sugar and water. Stir over medium heat until the sugar dissolves completely, about 3-4 minutes. Do not let it boil hard. Remove from heat and stir in the salt and vanilla extract. Let the simple syrup cool completely to room temperature.
If using fresh peaches, blanch them in boiling water for 30-45 seconds, then transfer to ice water. Peel the skin off starting from the scored X at the bottom. Halve each peach, remove the pit, and roughly chop the flesh. If using frozen peaches, thaw completely and drain any excess liquid.
Add the chopped peaches, cooled simple syrup, and fresh lemon juice to a blender. Blend on high speed for 60-90 seconds until completely smooth. For an ultra-silky texture, strain the puree through a fine mesh strainer. Taste and adjust sweetness or acidity if needed.
Transfer the peach puree to a bowl, cover tightly with plastic wrap pressed directly onto the surface, and refrigerate for at least 1 hour until thoroughly chilled.
Without an ice cream machine: Pour the chilled puree into a shallow freezer-safe container and freeze for 45 minutes. Scrape and stir vigorously with a fork, breaking up ice crystals. Repeat this process every 30-45 minutes for 3-4 hours. After the final stir, press plastic wrap onto the surface, seal, and freeze for 1-2 more hours until firm.
With an ice cream machine: Pour the chilled peach puree into your ice cream machine and churn according to manufacturer instructions, usually 20-25 minutes, until it reaches a soft-serve consistency. Transfer to a freezer-safe container, press plastic wrap onto the surface, seal, and freeze for 2-3 hours to firm up.
Remove the sorbet from the freezer 5-10 minutes before serving to soften. Scoop into chilled bowls or waffle cones and serve immediately. Enjoy!