Combine berries, water, sugar, and orange extract in a small saucepan and set over medium-high heat. Bring to a boil and boil for 1 minute, then cover and remove from heat. Let stand 30 minutes.
Set a food mill fitted with the fine disk over a medium bowl. Pour contents of saucepan into the food mill and turn crank to force through cranberries. Be sure to scrape any puree off the bottom of the disk into the bowl. Discard skins. (Alternatively, use a blender or food processor to puree cranberries and then force through a fine sieve to remove skins.)
Whisk in heavy cream, bourbon, allspice, and salt. Cover and let mixture chill for at least four hours.
Once chilled, pour into bowl of ice cream maker and chill according to manufacturer instructions.
Notes
Be sure to plan ahead for chill time, and have your ice cream bowl already frozen.
You can skip the bourbon if you prefer; this will make the ice cream somewhat harder for scooping.