Bring a large pot of water to a boil. Score an X on the bottom of each peach, then submerge them in boiling water for 30-45 seconds. Transfer immediately to an ice bath. Once cool enough to handle, peel the skins off. Slice the peaches and toss with lemon juice to prevent browning.
Place half the peaches in a food processor and pulse until smooth. Chop the remaining peaches into small chunks for texture variation.
In a medium saucepan, combine the milk and half the cream over medium heat until steaming but not boiling.
Meanwhile, whisk together egg yolks, sugar, and salt in a separate bowl until pale and slightly thickened.
Slowly pour 1 cup of the hot milk mixture into the egg mixture while continuously whisking. Then, pour the tempered egg mixture back into the saucepan with the remaining milk.
Cook the mixture over medium-low heat, stirring constantly with a wooden spoon until it reaches 170°F or coats the back of the spoon.
Strain the custard through a fine-mesh sieve into a bowl containing the remaining cream. Add vanilla and almond extracts, then fold in the peach purée and chunks.
Cover the mixture with plastic wrap, pressing it directly onto the surface to prevent a skin from forming. Refrigerate for at least 2 hours or overnight.
Pour the chilled mixture into your ice cream maker and churn according to manufacturer's instructions, usually 20-30 minutes. The ice cream is ready when it reaches the consistency of soft-serve.
Transfer to a freezer-safe container, cover with parchment paper directly on the surface, and freeze until firm, at least 4 hours or preferably overnight.