In a medium saucepan, combine the heavy cream, whole milk, granulated sugar, and salt. Heat over medium heat until the mixture is hot but not boiling, stirring occasionally to dissolve the sugar.
In a separate bowl, whisk the egg yolks until smooth.
Slowly pour about 1 cup of the hot cream mixture into the egg yolks while whisking constantly to temper the yolks.
Pour the tempered egg yolk mixture back into the saucepan with the remaining cream mixture.
Cook over medium heat, stirring constantly with a wooden spoon or spatula, until the mixture thickens enough to coat the back of the spoon (about 170°F or 77°C). Do not boil.
Remove from heat and whisk in the peanut butter and vanilla extract until fully combined and smooth.
Pour the mixture through a fine-mesh sieve into a clean bowl to remove any cooked egg bits.
Cover the bowl with plastic wrap, pressing it directly onto the surface of the custard to prevent a skin from forming. Refrigerate until completely chilled, at least 4 hours or overnight.
Once chilled, pour the custard into an ice cream maker and churn according to the manufacturer’s instructions until it reaches a soft-serve consistency.
Gently fold the chopped peanut butter cookies into the churned ice cream.
Transfer the ice cream to an airtight container and freeze for at least 2 hours to firm up before serving.