Place the peaches and honey in the jar of a blender. Puree until smooth. Set aside.
In a medium saucepan, bring the milk almost to a boil over medium heat. In a second medium saucepan, blend together the egg yolks, sugar and salt with a wooden spoon until thick and light (be careful not to make the mixture foamy). Slowly stir in half the hot milk, then transfer the mixture to the other saucepan with the remaining milk and stir until blended.
Decrease the heat to low and cook gently, stirring constantly with a wooden spoon. Continue stirring the custard until thick enough to coat the back of the spoon and the mixture reaches 175 degrees F on an instant-read thermometer. Remove from the heat.
Set a sieve over a large, clean bowl and pass the custard through the sieve. Discard any eggy solids. Add the peach puree and vanilla and stir to combine. Chill the custard completely over an ice bath made by filling a large bowl with ice cubes and water, or refrigerate it until chilled. Churn the ice cream according to the instructions for your ice-cream maker.