In a medium saucepan, combine the whole milk, heavy cream, granulated sugar, and salt. Heat over medium heat, stirring occasionally, until the sugar dissolves and the mixture is hot but not boiling, about 5 minutes.
In a separate bowl, whisk the egg yolks until smooth.
Slowly pour about 1 cup of the hot milk mixture into the egg yolks, whisking constantly to temper the eggs.
Pour the tempered egg yolk mixture back into the saucepan with the remaining milk mixture.
Cook over medium-low heat, stirring constantly with a wooden spoon or heatproof spatula, until the mixture thickens enough to coat the back of the spoon, about 5-7 minutes. Do not let it boil.
Remove the saucepan from heat and whisk in the peanut butter and vanilla extract until smooth and fully incorporated.
Pour the custard 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 the custard 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.
During the last 2 minutes of churning, add the chopped peanut butter cookies to incorporate evenly.
Transfer the ice cream to an airtight container and freeze for at least 2 hours to firm up before serving.