In a medium saucepan, combine the heavy cream, whole milk, granulated sugar, and salt. Heat over medium heat until the mixture is hot and the sugar has dissolved, about 5 minutes. Do not boil.
In a separate bowl, whisk the egg yolks until smooth.
Gradually pour about 1 cup of the hot cream mixture into the egg yolks, whisking constantly to temper the eggs.
Pour the tempered egg yolk mixture back into the saucepan with the remaining cream mixture.
Cook over medium-low heat, stirring constantly with a wooden spoon or spatula, until the mixture thickens and coats the back of the spoon, about 5 minutes. Do not let it boil.
Remove from heat and whisk in the peanut butter and vanilla extract until smooth and fully incorporated.
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, usually about 20-25 minutes.
During the last 5 minutes of churning, add the chopped peanut butter cookies to the ice cream maker to mix in evenly.
Transfer the ice cream to an airtight container and freeze for at least 2 hours to firm up before serving.