Bring a large saucepan of water to a boil. Meanwhile, prepare an ice-water bath; set aside. Place peaches in boiling water for 1 to 2 minutes; remove with a slotted spoon, and cool in ice-water bath.
Peel and pit peaches, reserving skin and pits. Slice each peach into 10 wedges, place in a nonreactive saucepan, and stir in lemon juice and 2 tablespoons sugar. Cook over medium-high heat, stirring occasionally, until peaches are tender and liquid has thickened slightly, 10 to 15 minutes. Let cool.
Stir peach schnapps into peach mixture. Transfer to the bowl of a food processor, and pulse to a chunky puree, 10 to 15 pulses; set aside.
In a large saucepan set over high heat, bring milk, cream, peach skins, and pits to a boil. Reduce heat to medium low; simmer 5 minutes. Cover, and let stand 15 minutes.
Meanwhile, in the bowl of an electric mixer fitted with the whisk attachment, combine egg yolks, salt, remaining sugar, and vanilla. Beat at medium-high speed until mixture is thick and pale yellow, about 3 minutes.
Strain milk mixture into egg mixture; discard solids. Beat at low speed to combine. Cook over low heat, stirring, until mixture coats the back of a wooden spoon, 3 to 5 minutes. Pass through a sieve into a bowl set in ice-water bath. When chilled, stir in peach puree. Freeze in an ice-cream maker according to manufacturer's instructions, in two batches if necessary. Store in an airtight plastic container in the freezer for up to 1 week.