Place shelled pistachios in a food processor and pulse with 1/4 cup of granulated sugar until fine. Gradually drizzle in 1/4 cup of whole milk while blending until smooth.

In a saucepan, heat remaining 1 3/4 cups of whole milk and 1 cup of heavy cream over medium heat until hot but not boiling.

Whisk together 4 egg yolks, 1/2 cup of granulated sugar, and a pinch of salt in a mixing bowl until pale and fluffy. Slowly add 1 cup of hot milk mixture while whisking continuously.

Return the tempered egg mixture to the saucepan with the remaining hot milk and cook over medium-low heat, stirring constantly until custard thickens.

Remove from heat, stir in the pistachio paste and vanilla extract, then strain the mixture into a bowl. Cover and refrigerate for at least 4 hours or overnight.

Pour chilled ice cream mixture into an ice cream maker and churn according to instructions for 20-25 minutes, or freeze in a shallow dish, stirring every 30 minutes for 3 hours.

Scoop ice cream into bowls or cones to serve immediately or store in a freezer-safe container for up to one week.