Soak pistachios in filtered water overnight. Drain and rinse. Optional: peel off the dark skins to get a better color and texture. This is a tedious step, and I don't think the end product would suffer too much if you skipped it.
Place the soaked pistachios and coconut milk in blender; process until smooth (taking breaks to scrape down the sides and blend again if you have a less-than-high-powered blender like mine).
Bring the pistachio milk, sugar, honey and corn syrup to a simmer in saucepan over medium heat. Remove from heat, stir in cardamom, vanilla extract, and cognac and set aside to cool. Strain through a sieve.
Cover and chill mixture in refrigerator for 2-3 hours or overnight.
Process ice cream as per manufacturer’s instructions. In the last minute of processing, add in chocolate. Cover and freeze.