Homemade ice cream works best when you give it time to chill between making the base and churning it, so factor that into your timing.
Make the Pistachio Paste
To make your own pistachio paste, toast the pistachios on a baking sheet at 325°F / 160°C for 5-7 minutes until fragrant, then blend them in a food processor until smooth. If you find it is too dry you can add a couple tablespoons of neutral oil. For a detailed step-by-step guide, see my homemade pistachio paste recipe. Alternatively, you can buy ready-made pistachio paste (see notes).
Make the Pistachio Ice Cream
Add the heavy cream, whole milk, corn syrup and salt to a saucepan over medium heat until scalding hot (steam rising and with small bubbles on the surface). Don't let it boil.
Whisk the egg yolks and sugar together until pale and slightly thickened, then slowly pour the hot milk mixture into the egg yolks, whisking as you go. It's important to do this gradually to avoid scrambling the eggs.
Pour the whole mixture back into the pan and heat, whisking constantly until it thickens enough to coat the back of a spoon and draw a line through it.
Whisk in the extracts and pistachio paste until incorporated. For the smoothest ice cream, you can use an immersion blender or high-speed blender here to emulsify it completely.
Cover the mixture with plastic wrap touching the surface and let it come to room temperature (or cool quickly in an ice bath). Chill in the refrigerator for at least 4 hours, preferably overnight. This will give the final ice cream a better flavor and texture.
When you're ready to churn, follow the instructions of your specific ice cream maker. Churn until the ice cream is gathering at the edges of the bowl, then use a spoon to check the consistency - it should be about as thick as soft serve ice cream, and should be able to hold its shape on a spoon.
Pour into an airtight container and freeze for a further 4 hours before scooping and serving.