In a medium-size saucepan, combine the peaches, lemon juice, a pinch salt, and ½ cup sugar. Let the mixture sit until the peaches soften slightly and a sweet syrupy liquid accumulates. This process should take about 1 hour.
Position sieve over medium bowl set in an ice-water bath; set aside. Heat milk, cream, and 1/2 cup sugar in medium-size heavy saucepan over medium heat, stirring occasionally, until steam appears, 5 to 6 minutes. Turn off heat.
Meanwhile, whisk yolks and remaining 6 tablespoons sugar in medium bowl until pale yellow. Stir half the warmed milk mixture into beaten yolk mixture until just blended.
Return milk-yolk mixture to the saucepan of the remaining warmed milk mixture. Heat milk-yolk mixture over medium-low heat, stirring constantly with a wooden spoon until steam appears. Continue stirring until the foam subsides and the custard just begins to thicken or an instant-read thermometer registers 180° F. The mixture must not boil or it will curdle.
Remove from heat and immediately strain the custard into the prepared bowl. Cool custard mixture to room temperature, stir in vanilla, then cover and refrigerate until an instant-read thermometer registers 40 degrees F, at least 2, and up to 24 hours.
COOK AND COOL PEACHES:
Meanwhile, heat softened peaches and their liquid, stirring occasionally, over medium-high heat until peaches are tender and flesh has broken down, 3 to 4 minutes. Transfer to bowl and refrigerate until cold, at least 4 and up to 24 hours.
CHURN ICE CREAM:
Strain chilled peaches, reserving liquid. Stir reserved peach liquid into chilled custard mixture; pour into ice cream machine canister and churn, following manufacturer’s instructions, until mixture is frozen and resembles soft-serve ice cream, 25 to 30 minutes. Add peaches; continue to churn until combined, about 30 seconds longer.
FREEZE AND SERVE:
Transfer ice cream to airtight container. Freeze until firm, about 2 hours.