Bring a medium pot of water to a boil. Meanwhile, lightly score bottom of each peach with an X, using a paring knife. Add peaches to boiling water and blanch until skin is pulling away, about 1 minute. Transfer to a bowl of ice water. When cool enough to handle, peel and pit peaches. Cut into 1/2-inch pieces.
Combine peaches and sugar in a medium saucepan. Cook over medium heat, stirring constantly, until just caramelized and syrupy, about 20 minutes. Let cool completely. (You should have about 1 1/4 cups.)
Chill a loaf pan in freezer at least 10 minutes. Freeze and churn custard base in an ice cream maker according to manufacturer's directions. Add peaches at end of churning and churn 5 more minutes; transfer to pan. Cover with plastic wrap and freeze until hardened, at least 2 hours; ice cream can be frozen up to 2 weeks ahead.