In the bowl of a stand mixer (or in a large bowl using a handheld mixer), add the ricotta, mascarpone, and sugar and mix on medium speed until just combined about 1 minute.
Add the heavy cream, milk, vanilla bean paste, and cinnamon. Mix on low speed until combined, 1-2 minutes. Fold in the cannoli shell pieces, chocolate chips, and pistachios. Cover the mixture and chill for 1 hour.
Remove the cannoli ice cream mixture from the refrigerator and freeze it in your ice cream maker as per the manufacturer's instructions. Scoop ice cream into containers and freeze until solid, about 3 hours.
Scoop and serve!
*This recipe makes 3-4 frozen pints of ice cream. *I usually get cannoli shells from Amazon or World Market. *If you can't find cannoli shells, then you can use 1/2 cup of broken waffle cone pieces.