Remove 2 Tablespoons of the ⅔ cup sugar and mix with 1 tsp xanthan gum. Set aside for later. Add the 4 egg yolks, milk, and remaining ⅔ cup sugar to the blender. Select the Mixer function. This will bring the temperature of the eggs to 160°F/71°C which is high enough to kill any salmonella, but not high enough to cook the eggs.
When the mixer function has completed it's cycle, add in the heavy cream, the vanilla extract and the vanilla paste. Add the sugar/xanthan gum to the blender along with the vodka if using. Select the ice cream function.
Pour the mixture into a large measuring cup or container and chill for about an hour. Pour into shallow containers and freeze for 6-8 hours.