1. Peel peaches if desired (see note above), then pit and slice them. In a food processor or blender, purée 2/3 of the peaches with 1 tablespoon lemon juice, vanilla, and 1/3 cup sugar. Pour into a large bowl, add half-and-half, and nest bowl in ice water; stir often until cold, 10 to 15 minutes. Or cover and chill until cold, about 1 hour.
2. In a bowl, with a potato masher or fork, coarsely mash remaining peaches with 1/2 tablespoon lemon juice and 1/3 cup sugar. Set container in ice water and stir often until cold, about 5 minutes. Add to half-and-half mixture; taste, and add more sugar if desired.
3. Pour chilled mixture into a 1-quart (or larger) ice cream maker. Freeze according to manufacturer's directions until mixture is firm enough to scoop, dasher is hard to turn, or machine stops.
4. Serve ice cream softly frozen, or package airtight and freeze up to 1 week; let stand at room temperature 10 to 20 minutes to soften for scooping.