In a food processor, pulse half the peaches with 1/3 cup sugar. Add the remaining peaches; pulse until almost smooth. Transfer to a large bowl.
In a medium bowl, whisk the egg yolks and the remaining 1/3 cup sugar until pale yellow, about 1 minute; set aside. In a medium saucepan, heat the half-and-half over medium heat until just simmering, about 5 minutes. Slowly whisk about one- quarter of the hot half-and-half into the yolk mixture.
Whisk the yolk mixture into the half-and-half in the saucepan and cook over medium heat, whisking constantly, until the mixture thickens enough to coat the back of a spoon, about 2 minutes. (Do not boil.)
Strain the warm egg mixture through a fine-mesh sieve into the peach puree. Stir to combine, then whisk in the almond extract. Set the bowl of custard in a large bowl of ice water and whisk until cold. Process the custard in an ice cream maker, adding the almonds when the ice cream is almost done processing, about 25 minutes. Serve with ginger cookies.